|
|
| Line 1: |
Line 1: |
| 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
| | by Nas, is very fitting and the film agrees with it. You may discover this probably the most time-consuming part of building a Word - Press MLM website. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. Word - Press also provides protection against spamming, as security is a measure issue. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br> |
| *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 + 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]].
| | Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. We also help to integrate various plug-ins to expand the functionalities of the web application. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. As of now, Pin2Press is getting ready to hit the market. If you have any inquiries concerning where and the best ways to utilize [http://l4.vc/wordpress_backup_plugin_556698 wordpress backup plugin], you could contact us at the web page. Use this section to change many formatting elements. These frequent updates have created menace in the task of optimization. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files. <br><br>A built-in widget which allows you to embed quickly video from popular websites. I didn't straight consider near it solon than one distance, I got the Popup Ascendancy plugin and it's up and lengthways, likely you make seen it today when you visited our blog, and I yet customize it to fit our Thesis Wound which gives it a rattling uncomparable visage and search than any different popup you know seen before on any added journal, I hump arrogated asset of one of it's quatern themes to make our own. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. If you are looking for Hire Wordpress Developer then just get in touch with him. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Customers within a few seconds after visiting a site form their opinion about the site. |
| | |
| ==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'' + 1,2''n'' + 1</sub>. This graph has an equitable 2-coloring, given by its bipartition. However, it does not have an equitable (2''n'' + 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'' + 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 + 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 {{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'' + 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'' + 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'';<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> + 1)(Δ<sub>2</sub> + 1) ≤ ''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 ≤ ''n''/3 − 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]]
| |
by Nas, is very fitting and the film agrees with it. You may discover this probably the most time-consuming part of building a Word - Press MLM website. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. Word - Press also provides protection against spamming, as security is a measure issue. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting.
Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. We also help to integrate various plug-ins to expand the functionalities of the web application. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.
Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. As of now, Pin2Press is getting ready to hit the market. If you have any inquiries concerning where and the best ways to utilize wordpress backup plugin, you could contact us at the web page. Use this section to change many formatting elements. These frequent updates have created menace in the task of optimization. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files.
A built-in widget which allows you to embed quickly video from popular websites. I didn't straight consider near it solon than one distance, I got the Popup Ascendancy plugin and it's up and lengthways, likely you make seen it today when you visited our blog, and I yet customize it to fit our Thesis Wound which gives it a rattling uncomparable visage and search than any different popup you know seen before on any added journal, I hump arrogated asset of one of it's quatern themes to make our own. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. If you are looking for Hire Wordpress Developer then just get in touch with him. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL.
He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Customers within a few seconds after visiting a site form their opinion about the site.