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In the mathematical field of graph theory, the Chvátal graph is an undirected graph with 12 vertices and 24 edges, discovered by . It is triangle-free: its girth (the length of its shortest cycle) is four. It is 4-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. This graph is not vertex-transitive: the automorphisms group has one orbit on vertices of size 8, and one of size 4. By Brooks’ theorem, every ''k''-regular graph (except for odd cycles and cliques) has chromatic number at most ''k''. It was also known since 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, 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 ), 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''. An alternative conjecture of 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 : 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. The Chvátal graph is Hamiltonian, and plays a key role in a proof by that it is NP-complete to determine whether a triangle-free Hamiltonian graph is 3-colorable. The characteristic polynomial of the Chvátal graph is . The Tutte polynomial of the Chvátal graph has been computed by . The independence number of this graph is 4. ==Gallery== File:Chvatal graph 4COL.svg|The chromatic number of the Chvátal graph is 4. File:chvatal graph 4color edge.svg|The chromatic index of the Chvátal graph is 4. File:Chvatal Lombardi.svg|The Chvátal graph is Hamiltonian. File:Chvátal graph.svg|Alternative drawing of the Chvátal graph. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Chvátal graph」の詳細全文を読む スポンサード リンク
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