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| | The name of the writer is Hipolito. To do archery is sole hobby her husband doesn't approve using. My house is now in Arizona and my parents live in the neighborhood. Filing is how she makes assets. See what's new on my website here: https://www.facebook.com/FamilyFarmSeasideHack<br><br>Feel free to surf to my web site [https://www.facebook.com/FamilyFarmSeasideHack Family farm seaside hack] |
| | name = Chvátal graph
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| | image = [[File:Chvatal graph.draw.svg|200px]]
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| | namesake = [[Václav Chvátal]]
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| | vertices = 12
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| | edges = 24
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| | chromatic_number = 4
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| | chromatic_index = 4
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| | girth = 4
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| | radius = 2
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| | diameter = 2
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| | automorphisms = 8 ([[Dihedral group|D<sub>4</sub>]])
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| | properties = [[regular graph|Regular]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[triangle-free graph|Triangle-free]]<br>[[Eulerian graph|Eulerian]]
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| }}
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| In the [[mathematics|mathematical]] field of [[graph theory]], the '''Chvátal graph''' is an [[undirected graph]] with 12 vertices and 24 edges, discovered by {{harvs|first=Václav|last=Chvátal|authorlink=Václav Chvátal|year=1970|txt}}.
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| It is [[triangle-free graph|triangle-free]]: its [[girth (graph theory)|girth]] (the length of its shortest cycle) is four. It is 4-[[regular graph|regular]]: each vertex has exactly four neighbors. And its [[chromatic number]] is 4: it can be colored using four colors, but not using only three. It is, as Chvátal observes, the smallest possible 4-chromatic 4-regular triangle-free graph; the only smaller 4-chromatic triangle-free graph is the [[Grötzsch graph]], which has 11 vertices but has maximum degree 5 and is not regular.
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| This graph is not [[vertex-transitive graph|vertex-transitive]]: the automorphisms group has one orbit on vertices of size 8, and one of size 4.
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| By [[Brooks’ theorem]], every ''k''-regular graph (except for odd cycles and cliques) has chromatic number at most ''k''. It was also known since {{harvtxt|Erdős|1959}} that, for every ''k'' and ''l'' there exist ''k''-chromatic graphs with girth ''l''. In connection with these two results and several examples including the Chvátal graph,
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| {{harvs|first=Branko|last=Grünbaum|authorlink=Branko Grünbaum|year=1970|txt}} conjectured that for every ''k'' and ''l'' there exist ''k''-chromatic ''k''-regular graphs with girth ''l''. The Chvátal graph solves the case ''k'' = ''l'' = 4 of this conjecture. Grünbaum's conjecture was disproven for sufficiently large ''k'' by Johannsen (see {{harvnb|Reed|1998}}), who showed that the chromatic number of a triangle-free graph is O(Δ/log Δ) where Δ is the maximum vertex degree and the O introduces [[big O notation]]. However, despite this disproof, it remains of interest to find examples such as the Chvátal graph of high-girth ''k''-chromatic ''k''-regular graphs for small values of ''k''.
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| An alternative conjecture of {{harvs|first=Bruce|last=Reed|authorlink=Bruce Reed (mathematician)|year=1998|txt}} states that high-degree triangle-free graphs must have significantly smaller chromatic number than their degree, and more generally that a graph with maximum degree Δ and [[maximum clique]] size ω must have chromatic number
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| :<math>\chi(G)\le\left\lceil\frac{\Delta+\omega+1}{2}\right\rceil.</math>
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| The case ω = 2 of this conjecture follows, for sufficiently large Δ, from Johanssen's result. The Chvátal graph shows that the rounding up in Reed's conjecture is necessary, because for the Chvátal graph, (Δ + ω + 1)/2 = 7/2, a number that is less than the chromatic number but that becomes equal to the chromatic number when rounded up.
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| The Chvátal graph is [[Hamiltonian cycle|Hamiltonian]], and plays a key role in a proof by {{harvtxt|Fleischner|Sabidussi|2002}} that it is [[NP-complete]] to determine whether a triangle-free Hamiltonian graph is 3-colorable.
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| The [[characteristic polynomial]] of the Chvátal graph is <math>(x-4) (x-1)^4 x^2 (x+1) (x+3)^2 (x^2+x-4)</math>. The [[Tutte polynomial]] of the Chvátal graph has been computed by {{harvtxt|Björklund|Husfeldt|Kaski|Koivisto|2008}}.
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| The [[independence number]] of this graph is 4.
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| ==Gallery==
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| <gallery>
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| File:Chvatal graph 4COL.svg|The [[chromatic number]] of the Chvátal graph is 4.
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| File:chvatal graph 4color edge.svg|The [[chromatic index]] of the Chvátal graph is 4.
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| File:Chvatal Lombardi.svg|The Chvátal graph is [[Hamiltonian graph|Hamiltonian]].
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| File:Chvátal graph.svg|Alternative drawing of the Chvátal graph.
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| </gallery>
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| ==References==
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| *{{citation
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| | last1 = Björklund | first1 = Andreas
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| | last2 = Husfeldt | first2 = Thore
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| | last3 = Kaski | first3 = Petteri
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| | last4 = Koivisto | first4 = Mikko
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| | contribution = Computing the Tutte Polynomial in Vertex-Exponential Time
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| | doi = 10.1109/FOCS.2008.40
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| | isbn = 978-0-7695-3436-7
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| | location = Washington, DC, USA
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| | pages = 677–686
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| | publisher = IEEE Computer Society
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| | title = FOCS '08: Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
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| | year = 2008
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| | arxiv = 0711.2585}}.
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| *{{citation
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| | last = Chvátal | first = V. | author-link = Václav Chvátal
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| | doi = 10.1016/S0021-9800(70)80057-6
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| | issue = 1
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| | journal = Journal of Combinatorial Theory
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| | pages = 93–94
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| | title = The smallest triangle-free 4-chromatic 4-regular graph
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| | volume = 9
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| | year = 1970}}.
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| *{{citation
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| | doi = 10.4153/CJM-1959-003-9
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| | last = Erdős | first = Paul | author-link = Paul Erdős
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| | journal = Canadian Journal of Mathematics
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| | pages = 34–38
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| | title = Graph theory and probability
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| | volume = 11
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| | year = 1959}}.
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| *{{citation
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| | author2-link = Gert Sabidussi
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| | last1 = Fleischner | first1 = Herbert
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| | last2 = Sabidussi | first2 = Gert
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| | doi = 10.1002/jgt.10079
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| | issue = 2
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| | journal = Journal of Graph Theory
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| | pages = 125–140
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| | title = 3-colorability of 4-regular Hamiltonian graphs
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| | volume = 42
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| | year = 2002}}.
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| *{{citation
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| | last = Grünbaum | first = B. | author-link = Branko Grünbaum
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| | doi = 10.2307/2316101
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| | issue = 10
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| | journal = American Mathematical Monthly
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| | pages = 1088–1092
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| | title = A problem in graph coloring
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| | volume = 77
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| | year = 1970
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| | publisher = Mathematical Association of America
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| | jstor = 2316101}}.
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| *{{citation
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| | last = Reed | first = B. A. | authorlink = Bruce Reed (mathematician)
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| | doi = 10.1002/(SICI)1097-0118(199804)27:4<177::AID-JGT1>3.0.CO;2-K
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| | issue = 4
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| | journal = Journal of Graph Theory
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| | pages = 177–212
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| | title = ω, Δ, and χ
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| | volume = 27
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| | year = 1998}}.
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| ==External links==
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| *{{mathworld|title=Chvátal Graph|urlname=ChvatalGraph}}
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| {{DEFAULTSORT:Chvatal graph}}
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| [[Category:Individual graphs]]
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| [[Category:Regular graphs]]
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| [[Category:4-chromatic graphs]]
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The name of the writer is Hipolito. To do archery is sole hobby her husband doesn't approve using. My house is now in Arizona and my parents live in the neighborhood. Filing is how she makes assets. See what's new on my website here: https://www.facebook.com/FamilyFarmSeasideHack
Feel free to surf to my web site Family farm seaside hack