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In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that

No two adjacent vertices have the same color, and
The numbers of vertices in any two color classes differ by at most one.

That is, the partition of vertices among the different colors is as uniform as possible. For instance, giving each vertex a distinct color would be equitable, but would typically use many more colors than are necessary in an optimal equitable coloring. An equivalent way of defining an equitable coloring is that it is an embedding of the given graph as a subgraph of a Turán graph. There are two kinds of chromatic number associated with equitable coloring.^[1] The equitable chromatic number of a graph G is the smallest number k such that G has an equitable coloring with k colors. But G might not have equitable colorings for some larger numbers of colors; the equitable chromatic threshold of G is the smallest k such that G has equitable colorings for any number of colors greater than or equal to k.^[2]

The Hajnal–Szemerédi theorem, posed as a conjecture by Template:Harvs and proven by Template:Harvs, states that any graph with maximum degree Δ has an equitable coloring with Δ + 1 colors. Several related conjectures remain open. Polynomial time algorithms are also known for finding a coloring matching this bound, and for finding optimal colorings of special classes of graphs, but the more general problem of finding an equitable coloring of an arbitrary graph with a given number of colors is NP-complete.

Examples

An equitable coloring of the star K_1,5.

The star K_1,5 shown in the illustration is a complete bipartite graph, and therefore may be colored with two colors. However, the resulting coloring has one vertex in one color class and five in another, and is therefore not equitable. The smallest number of colors in an equitable coloring of this graph is four, as shown in the illustration: the central vertex must be the only vertex in its color class, so the other five vertices must be split among at least three color classes in order to ensure that the other color classes all have at most two vertices. More generally, Template:Harvtxt observes that any star K_1,n needs $\scriptstyle 1+\lceil n/2\rceil$ colors in any equitable coloring; thus, the chromatic number of a graph may differ from its equitable coloring number by a factor as large as n/4. Because K_1,5 has maximum degree five, the number of colors guaranteed for it by the Hajnal–Szemerédi theorem is six, achieved by giving each vertex a distinct color.

Another interesting phenomenon is exhibited by a different complete bipartite graph, K_{2n + 1,2n + 1}. This graph has an equitable 2-coloring, given by its bipartition. However, it does not have an equitable (2n + 1)-coloring: any equitable partition of the vertices into that many color classes would have to have exactly two vertices per class, but the two sides of the bipartition cannot each be partitioned into pairs because they have an odd number of vertices. Therefore, the equitable chromatic threshold of this graph is 2n + 2, significantly greater than its equitable chromatic number of two.

Hajnal–Szemerédi theorem

Brooks' theorem states that any connected graph with maximum degree Δ has a Δ-coloring, with two exceptions (complete graphs and odd cycles). However, this coloring may in general be far from equitable. Template:Harvs conjectured that an equitable coloring is possible with only one more color: any graph with maximum degree Δ has an equitable coloring with Δ + 1 colors. The case Δ = 2 is straightforward (any union of paths and cycles may be equitably colored by using a repeated pattern of three colors, with minor adjustments to the repetition when closing a cycle) and the case Δ + 1= n/3 had previously been solved by Template:Harvtxt. The full conjecture was proven by Template:Harvtxt, and is now known as the Hajnal–Szemerédi theorem. Their original proof was long and complicated; a simpler proof was given by Template:Harvtxt. A polynomial time algorithm for finding equitable colorings with this many colors was described by Kierstead and Kostochka; they credit Marcelo Mydlarz and Endre Szemerédi with a prior unpublished polynomial time algorithm. Kierstead and Kostochka also announce but do not prove a strengthening of the theorem, to show that an equitable k-coloring exists whenever every two adjacent vertices have degrees adding to at most 2k + 1.

Template:Harvtxt conjectured a form of Brooks' theorem for equitable coloring: every connected graph with maximum degree Δ has an equitable coloring with Δ or fewer colors, with the exceptions of complete graphs and odd cycles. A strengthened version of the conjecture states that each such graph has an equitable coloring with exactly Δ colors, with one additional exception, a complete bipartite graph in which both sides of the bipartition have the same odd number of vertices.^[1]

Template:Harvtxt proposed a strengthening of the Hajnal–Szemerédi theorem that also subsumes Dirac's theorem that dense graphs are Hamiltonian: he conjectured that, if every vertex in an n-vertex graph has at least kn/(k + 1) neighbors, then the graph contains as a subgraph the graph formed by connecting vertices that are at most k steps apart in an n-cycle. The case k = 1 is Dirac's theorem itself. The Hajnal–Szemerédi theorem may be recovered from this conjecture by applying the conjecture for larger values of k to the complement graph of a given graph, and using as color classes contiguous subsequences of vertices from the n-cycle. Seymour's conjecture has been proven for graph in which n is sufficiently large relative to k;^[3] the proof uses several deep tools including the Hajnal–Szemerédi theorem itself.

Yet another generalization of the Hajnal–Szemerédi theorem is the Bollobás–Eldridge–Catlin conjecture (or BEC-conjecture for short).^[4] This states that if G₁ and G₂ are graphs on n vertices with maximum degree Δ₁ and Δ₂ respectively, and if (Δ₁ + 1)(Δ₂ + 1) ≤ n+1, then G₁ and G₂ can be packed. That is, G₁ and G₂ can be represented on the same set of n vertices with no edges in common. The Hajnal–Szemerédi theorem is the special case of this conjecture in which G₂ is a disjoint union of cliques. Template:Harvtxt provides a similar but stronger condition on Δ₁ and Δ₂ under which such a packing can be guaranteed to exist.

Special classes of graphs

For any tree with maximum degree Δ, the equitable chromatic number is at most

1+\lceil \Delta /2\rceil

^[5]

with the worst case occurring for a star. However, most trees have significantly smaller equitable chromatic number: if a tree with n vertices has Δ ≤ n/3 − O(1), then it has an equitable coloring with only three colors.^[6] Template:Harvtxt studies the equitable chromatic number of graph products.

Computational complexity

The problem of finding equitable colorings with as few colorings as possible (below the Hajnal-Szemerédi bound) has also been studied. A straightforward reduction from graph coloring to equitable coloring may be proven by adding sufficiently many isolated vertices to a graph, showing that it is NP-complete to test whether a graph has an equitable coloring with a given number of colors (greater than two). However, the problem becomes more interesting when restricted to special classes of graphs or from the point of view of parameterized complexity. Template:Harvtxt showed that, given a graph G and a number c of colors, it is possible to test whether G admits an equitable c-coloring in time O(n^O(t)), where t is the treewidth of G; in particular, equitable coloring may be solved optimally in polynomial time for trees (previously known due to Template:Harvnb) and outerplanar graphs. A polynomial time algorithm is also known for equitable coloring of split graphs.^[7] However, Template:Harvtxt prove that, when the treewidth is a parameter to the algorithm, the problem is W[1]-hard. Thus, it is unlikely that there exists a polynomial time algorithm independent of this parameter, or even that the dependence on the parameter may be moved out of the exponent in the formula for the running time.

Applications

One motivation for equitable coloring suggested by Template:Harvtxt concerns scheduling problems. In this application, the vertices of a graph represent a collection of tasks to be performed, and an edge connects two tasks that should not be performed at the same time. A coloring of this graph represents a partition of the tasks into subsets that may be performed simultaneously; thus, the number of colors in the coloring corresponds to the number of time steps required to perform the entire task. Due to load balancing considerations, it is desirable to perform equal or nearly-equal numbers of tasks in each time step, and this balancing is exactly what an equitable coloring achieves. Template:Harvtxt mentions a specific application of this type of scheduling problem, assigning university courses to time slots in a way that spreads the courses evenly among the available time slots and avoids scheduling incompatible pairs of courses at the same time as each other.

The Hajnal-Szemerédi theorem has also been used to bound the variance of sums of random variables with limited dependence (Template:Harvnb; Template:Harvnb). If (as in the setup for the Lovász local lemma) each variable depends on at most Δ others, an equitable coloring of the dependence graph may be used to partition the variables into independent subsets within which Chernoff bounds may be calculated, resulting in tighter overall bounds on the variance than if the partition were performed in a non-equitable way.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

Template:Refend

External links

ECOPT A Branch and Cut algorithm for solving the Equitable Coloring Problem

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+In [[graph theory]], an area of mathematics, an '''equitable coloring''' is an assignment of [[graph coloring|colors]] to the [[vertex (graph theory)|vertices]] of an [[undirected graph]], in such a way that
+*No two adjacent vertices have the same color, and
+*The numbers of vertices in any two color classes differ by at most one.
+That is, the partition of vertices among the different colors is as uniform as possible. For instance, giving each vertex a distinct color would be equitable, but would typically use many more colors than are necessary in an optimal equitable coloring. An equivalent way of defining an equitable coloring is that it is an embedding of the given graph as a [[Glossary of graph theory#Subgraphs|subgraph]] of a [[Turán graph]]. There are two kinds of [[chromatic number]] associated with equitable coloring.<ref name="F06">{{harvtxt|Furmańczyk|2006}}.</ref> The '''equitable chromatic number''' of a graph ''G'' is the smallest number ''k'' such that ''G'' has an equitable coloring with ''k'' colors. But ''G'' might not have equitable colorings for some larger numbers of colors; the '''equitable chromatic threshold''' of ''G'' is the smallest ''k'' such that ''G'' has equitable colorings for any number of colors greater than or equal to ''k''.<ref>Note that, when ''k'' is greater than the number of vertices in the graph, there nevertheless exists an equitable coloring with ''k'' colors in which all color classes have zero or one vertices in them, so every graph has an equitable chromatic threshold.</ref>
+The '''Hajnal–Szemerédi theorem''', posed as a conjecture by {{harvs|first=Paul|last=Erdős|authorlink=Paul Erdős|txt|year=1964}} and proven by {{harvs|first1=András|last1=Hajnal|author1-link=András Hajnal|first2=Endre|last2=Szemerédi|author2-link=Endre Szemerédi|year=1970|txt}}, states that any graph with maximum degree Δ has an equitable coloring with Δ&nbsp;+&nbsp;1 colors. Several related conjectures remain open. Polynomial time algorithms are also known for finding a coloring matching this bound, and for finding optimal colorings of special classes of graphs, but the more general problem of finding an equitable coloring of an arbitrary graph with a given number of colors is [[NP-complete]].
+==Examples==
+[[File:Equitable-K15.svg|thumb|An equitable coloring of the [[star (graph theory)|star]] ''K''<sub>1,5</sub>.]]
+The [[star (graph theory)|star]] ''K''<sub>1,5</sub> shown in the illustration is a [[complete bipartite graph]], and therefore may be colored with two colors. However, the resulting coloring has one vertex in one color class and five in another, and is therefore not equitable. The smallest number of colors in an equitable coloring of this graph is four, as shown in the illustration: the central vertex must be the only vertex in its color class, so the other five vertices must be split among at least three color classes in order to ensure that the other color classes all have at most two vertices. More generally, {{harvtxt|Meyer|1973}} observes that any star ''K''<sub>1,''n''</sub> needs <math>\scriptstyle 1+\lceil n/2\rceil</math> colors in any equitable coloring; thus, the chromatic number of a graph may differ from its equitable coloring number by a factor as large as ''n''/4. Because ''K''<sub>1,5</sub> has maximum degree five, the number of colors guaranteed for it by the Hajnal–Szemerédi theorem is six, achieved by giving each vertex a distinct color.
+Another interesting phenomenon is exhibited by a different complete bipartite graph, ''K''<sub>2''n''&nbsp;+&nbsp;1,2''n''&nbsp;+&nbsp;1</sub>. This graph has an equitable 2-coloring, given by its bipartition. However, it does not have an equitable (2''n''&nbsp;+&nbsp;1)-coloring: any equitable partition of the vertices into that many color classes would have to have exactly two vertices per class, but the two sides of the bipartition cannot each be partitioned into pairs because they have an odd number of vertices. Therefore, the equitable chromatic threshold of this graph is 2''n''&nbsp;+&nbsp;2, significantly greater than its equitable chromatic number of two.
+==Hajnal–Szemerédi theorem==
+[[Brooks' theorem]] states that any connected graph with maximum degree Δ has a Δ-coloring, with two exceptions ([[complete graph]]s and odd cycles). However, this coloring may in general be far from equitable. {{harvs|first=Paul|last=Erdős|authorlink=Paul Erdős|txt|year=1964}} [[Erdős conjecture|conjectured]] that an equitable coloring is possible with only one more color: any graph with maximum degree Δ has an equitable coloring with Δ&nbsp;+&nbsp;1 colors. The case Δ&nbsp;=&nbsp;2 is straightforward (any union of paths and cycles may be equitably colored by using a repeated pattern of three colors, with minor adjustments to the repetition when closing a cycle) and the case Δ&nbsp;+&nbsp;1=&nbsp;''n''/3 had previously been solved by {{harvtxt|Corrádi|Hajnal|1963}}. The full conjecture was proven by {{harvtxt|Hajnal|Szemerédi|1970}}, and is now known as the Hajnal–Szemerédi theorem. Their original proof was long and complicated; a simpler proof was given by {{harvtxt|Kierstead|Kostochka|2008}}. A polynomial time algorithm for finding equitable colorings with this many colors was described by Kierstead and Kostochka; they credit Marcelo Mydlarz and Endre Szemerédi with a prior unpublished polynomial time algorithm. Kierstead and Kostochka also announce but do not prove a strengthening of the theorem, to show that an equitable ''k''-coloring exists whenever every two adjacent vertices have degrees adding to at most 2''k''&nbsp;+&nbsp;1.
+{{harvtxt|Meyer|1973}} conjectured a form of Brooks' theorem for equitable coloring: every connected graph with maximum degree Δ has an equitable coloring with Δ or fewer colors, with the exceptions of complete graphs and odd cycles. A strengthened version of the conjecture states that each such graph has an equitable coloring with exactly Δ colors, with one additional exception, a [[complete bipartite graph]] in which both sides of the bipartition have the same odd number of vertices.<ref name="F06"/>
+{{harvtxt|Seymour|1974}} proposed a strengthening of the Hajnal–Szemerédi theorem that also subsumes [[Gabriel Andrew Dirac|Dirac]]'s theorem that [[dense graph]]s are [[Hamiltonian path|Hamiltonian]]: he conjectured that, if every vertex in an ''n''-vertex graph has at least ''kn''/(''k''&nbsp;+&nbsp;1) neighbors, then the graph contains as a subgraph the graph formed by connecting vertices that are at most ''k'' steps apart in an ''n''-cycle. The case ''k''&nbsp;=&nbsp;1 is Dirac's theorem itself. The Hajnal–Szemerédi theorem may be recovered from this conjecture by applying the conjecture for larger values of ''k'' to the [[complement graph]] of a given graph, and using as color classes contiguous subsequences of vertices from the ''n''-cycle. Seymour's conjecture has been proven for graph in which ''n'' is sufficiently large relative to ''k'';<ref>{{harvtxt|Komlós|Sárközy|Szemerédi|1998}}.</ref> the proof uses several deep tools including the Hajnal–Szemerédi theorem itself.
+Yet another generalization of the Hajnal–Szemerédi theorem is the Bollobás–Eldridge–Catlin conjecture (or BEC-conjecture for short).<ref>{{harvtxt|Bollobás|Eldridge|1978}}.</ref> This states that if ''G''<sub>1</sub> and ''G''<sub>2</sub> are graphs on ''n'' vertices with maximum degree Δ<sub>1</sub> and Δ<sub>2</sub> respectively, and if (Δ<sub>1</sub>&nbsp;+&nbsp;1)(Δ<sub>2</sub>&nbsp;+&nbsp;1)&nbsp;≤&nbsp;''n+1'', then ''G''<sub>1</sub> and ''G''<sub>2</sub> can be packed. That is, ''G''<sub>1</sub> and ''G''<sub>2</sub> can be represented on the same set of ''n'' vertices with no edges in common. The Hajnal–Szemerédi theorem is the special case of this conjecture in which ''G''<sub>2</sub> is a disjoint union of [[clique (graph theory)|cliques]]. {{harvtxt|Catlin|1974}} provides a similar but stronger condition on Δ<sub>1</sub> and Δ<sub>2</sub> under which such a packing can be guaranteed to exist.
+==Special classes of graphs==
+For any tree with maximum degree Δ, the equitable chromatic number is at most
+:<math>1+\lceil\Delta/2\rceil</math><ref>{{harvtxt|Meyer|1973}}.</ref>
+with the worst case occurring for a star. However, most trees have significantly smaller equitable chromatic number: if a tree with ''n'' vertices has Δ&nbsp;≤&nbsp;''n''/3&nbsp;&minus;&nbsp;O(1), then it has an equitable coloring with only three colors.<ref>{{harvtxt|Bodlaender|Guy|1983}}.</ref> {{harvtxt|Furmańczyk|2006}} studies the equitable chromatic number of [[graph products]].
+==Computational complexity==
+The problem of finding equitable colorings with as few colorings as possible (below the Hajnal-Szemerédi bound) has also been studied. A straightforward reduction from [[graph coloring]] to equitable coloring may be proven by adding sufficiently many isolated vertices to a graph, showing that it is [[NP-complete]] to test whether a graph has an equitable coloring with a given number of colors (greater than two). However, the problem becomes more interesting when restricted to special classes of graphs or from the point of view of [[parameterized complexity]]. {{harvtxt|Bodlaender|Fomin|2005}} showed that, given a graph ''G'' and a number ''c'' of colors, it is possible to test whether ''G'' admits an equitable ''c''-coloring in time O(''n''<sup>O(''t'')</sup>), where ''t'' is the [[treewidth]] of ''G''; in particular, equitable coloring may be solved optimally in polynomial time for [[tree (graph theory)|trees]] (previously known due to {{harvnb|Chen|Lih|1994}}) and [[outerplanar graph]]s. A polynomial time algorithm is also known for equitable coloring of [[split graph]]s.<ref>{{harvtxt|Chen|Ko|Lih|1996}}.</ref> However, {{harvtxt|Fellows|Fomin|Lokshtanov|Rosamond|2007}} prove that, when the treewidth is a parameter to the algorithm, the problem is W[1]-hard. Thus, it is unlikely that there exists a polynomial time algorithm independent of this parameter, or even that the dependence on the parameter may be moved out of the exponent in the formula for the running time.
+==Applications==
+One motivation for equitable coloring suggested by {{harvtxt|Meyer|1973}} concerns [[Job Shop Scheduling|scheduling]] problems. In this application, the vertices of a graph represent a collection of tasks to be performed, and an edge connects two tasks that should not be performed at the same time. A coloring of this graph represents a partition of the tasks into subsets that may be performed simultaneously; thus, the number of colors in the coloring corresponds to the number of time steps required to perform the entire task. Due to [[load balancing (computing)|load balancing]] considerations, it is desirable to perform equal or nearly-equal numbers of tasks in each time step, and this balancing is exactly what an equitable coloring achieves. {{harvtxt|Furmańczyk|2006}} mentions a specific application of this type of scheduling problem, assigning university courses to time slots in a way that spreads the courses evenly among the available time slots and avoids scheduling incompatible pairs of courses at the same time as each other.
+The Hajnal-Szemerédi theorem has also been used to bound the [[variance]] of sums of  random variables with limited dependence ({{harvnb|Pemmaraju|2001}}; {{harvnb|Janson|Ruciński|2002}}). If (as in the setup for the [[Lovász local lemma]]) each variable depends on at most Δ others, an equitable coloring of the dependence graph may be used to partition the variables into independent subsets within which [[Chernoff bound]]s may be calculated, resulting in tighter overall bounds on the variance than if the partition were performed in a non-equitable way.
+==Notes==
+{{reflist|2}}
+==References==
+{{refbegin|colwidth=30em}}
+*{{citation
+ | last1 = Bodlaender | first1 = Hans L. | author1-link = Hans L. Bodlaender
+ | last2 = Fomin | first2 = Fedor V.
+ | doi = 10.1016/j.tcs.2005.09.027
+ | mr = 2183465
+ | issue = 1
+ | journal = Theoretical Computer Science
+ | pages = 22–30
+ | title = Equitable colorings of bounded treewidth graphs
+ | volume = 349
+ | year = 2005}}.
+*{{citation | last1 = Bollobás | first1 = B. | last2= Eldridge | first2= S. E. | author1-link = Béla Bollobás | journal = [[Journal of Combinatorial Theory]] | series = Series B | pages = 105–124 | title = Packings of graphs and applications to computational complexity | volume = 25 | year = 1978 | doi = 10.1016/0097-3165(78)90073-0 | mr = 0511983}}.
+*{{citation
+ | last1 = Bollobás | first1 = Béla | author1-link = Béla Bollobás
+ | last2 = Guy | first2 = Richard K. | author2-link = Richard K. Guy
+ | doi = 10.1016/0095-8956(83)90017-5
+ | mr = 703602
+ | issue = 2
+ | journal = [[Journal of Combinatorial Theory]] | series = Series B
+ | pages = 177–186
+ | title = Equitable and proportional coloring of trees
+ | volume = 34
+ | year = 1983}}.
+*{{citation | last = Catlin | first =  Paul A. | journal = Discrete Math. | pages = 225–233 | title = Subgraphs of graphs. I. | volume = 10 | year = 1974 | doi = 10.1016/0012-365X(74)90119-8 | mr = 0357230}}
+*{{citation
+ | last1 = Chen | first1 = Bor-Liang
+ | last2 = Lih | first2 = Ko-Wei
+ | doi = 10.1006/jctb.1994.1032
+ | mr = 1275266
+ | issue = 1
+ | journal = [[Journal of Combinatorial Theory]] | series = Series B
+ | pages = 83–87
+ | title = Equitable coloring of trees
+ | volume = 61
+ | year = 1994}}.
+*{{citation
+ | last1 = Chen | first1 = Bor-Liang
+ | last2 = Ko | first2 = Ming-Tat
+ | last3 = Lih | first3 = Ko-Wei
+ | contribution = Equitable and ''m''-bounded coloring of split graphs
+ | mr = 1448914
+ | location = Berlin
+ | pages = 1–5
+ | publisher = Springer-Verlag
+ | series = Lecture Notes in Computer Science
+ | title = Combinatorics and Computer Science (Brest, 1995)
+ | volume = 1120
+ | year = 1996}}
+*{{citation | doi = 10.1007/BF01895727 | last1 = Corrádi | first1 = K. | last2 = Hajnal | first2 = A. | author2-link = András Hajnal | mr = 0200185 | journal = Acta Mathematica Academiae Scientiarum Hungaricae | pages = 423–439 | title = On the maximal number of independent circuits in a graph | volume = 14 | year = 1963}}.
+*{{citation | last = Erdős | first = Paul | authorlink = Paul Erdős | contribution = Problem 9 | title = Theory of Graphs and its Applications | editor-first = M. | editor-last = Fieldler | page = 159 | publisher = Czech Acad. Sci. Publ. | location = Prague | year = 1964}}.
+*{{citation
+ | last1 = Fellows | first1 = Michael | author1-link = Michael Fellows
+ | last2 = Fomin | first2 = Fedor V.
+ | last3 = Lokshtanov | first3 = Daniel
+ | last4 = Rosamond | first4 = Frances
+ | last5 = Saurabh | first5 = Saket
+ | last6 = Szeider | first6 = Stefan
+ | last7 = Thomassen | first7 = Carsten | author7-link = Carsten Thomassen
+ | contribution = On the complexity of some colorful problems parameterized by treewidth
+ | doi = 10.1007/978-3-540-73556-4_38
+ | mr = 2397574
+ | location = Berlin
+ | pages = 366–377
+ | publisher = Springer-Verlag
+ | series = Lecture Notes in Computer Science
+ | title = Combinatorial optimization and applications
+ | url = http://www.ii.uib.no/~daniello/papers/COCOA-coloring.pdf
+ | volume = 4616
+ | year = 2007}}.
+*{{citation
+ | last = Furmańczyk | first = Hanna
+ | mr = 2272280
+ | issue = 1
+ | journal = [[Opuscula Mathematica]]
+ | pages = 31–44
+ | title = Equitable coloring of graph products
+ | url = http://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2603.pdf
+ | volume = 26
+ | year = 2006}}.
+*{{citation|contribution=Proof of a conjecture of P. Erdős|first1=A.|last1=Hajnal|author1-link=András Hajnal|first2=E.|last2=Szemerédi|author2-link=Endre Szemerédi|title=Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969)|publisher=North-Holland|year=1970|pages=601–623|mr=0297607}}.
+*{{citation | last1 = Janson | first1 = Svante | authorlink=Svante Janson | last2 = Ruciński | first2 = Andrzej | doi = 10.1002/rsa.10031 | mr = 1900611 | issue = 3 | journal = Random Structures &Algorithms | pages = 317–342 | title = The infamous upper tail | volume = 20 | year = 2002}}.
+*{{citation | last1 = Kierstead | first1 = H. A. | last2 = Kostochka | first2 = A. V. | doi = 10.1017/S0963548307008619 | mr = 2396352 | issue = 2 | journal = Combinatorics, Probability and Computing | pages = 265–270 | title = A short proof of the Hajnal-Szemerédi theorem on equitable colouring | volume = 17 | year = 2008}}
+*{{citation | last1 = Komlós | first1 = János | author1-link= János Komlós (mathematician)| last2 = Sárközy | first2 = Gábor N. | last3 = Szemerédi | first3 = Endre | author3-link = Endre Szemerédi | doi = 10.1007/BF01626028 | mr = 1682919 | issue = 1 | journal = Annals of Combinatorics | pages = 43–60 | title = Proof of the Seymour conjecture for large graphs | volume = 2 | year = 1998}}.
+*{{citation|doi=10.2307/2319405|first=Walter|last=Meyer|title=Equitable coloring|journal=[[American Mathematical Monthly]]|volume=80|issue=8|year=1973|pages=920–922|publisher=Mathematical Association of America|jstor=2319405}}.
+*{{citation | last = Pemmaraju | first = Sriram V. | contribution = Equitable colorings extend Chernoff-Hoeffding bounds | pages = 924–925 | title = Proc. 12th ACM-SIAM Symp. Discrete Algorithms (SODA 2001) | url = http://portal.acm.org/citation.cfm?id=365411.365811 | year = 2001}}.
+*{{citation|first=P.|last=Seymour|authorlink=Paul Seymour (mathematician)|contribution=Problem section|title=Combinatorics: Proceedings of the British Combinatorial Conference 1973|editor1-first=T. P.|editor1-last=McDonough|editor2-first=V. C.|editor2-last=Mavron, Eds.|pages=201–202|publisher=Cambridge Univ. Press|location=Cambridge, UK|year=1974}}.
+{{refend}}
+== External links ==
+* [http://www.fceia.unr.edu.ar/~daniel/ecopt/index.html ''ECOPT''] A Branch and Cut algorithm for solving the Equitable Coloring Problem
+[[Category:Graph coloring]]

Transition path sampling: Difference between revisions

Revision as of 02:41, 15 March 2013

Contents

Examples

Hajnal–Szemerédi theorem

Special classes of graphs

Computational complexity

Applications

Notes

References

External links

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Transition path sampling: Difference between revisions

Revision as of 02:41, 15 March 2013

Examples

Hajnal–Szemerédi theorem

Special classes of graphs

Computational complexity

Applications

Notes

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