Chvátal graph

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Chvátal graph
Chvatal graph.draw.svg
Named after Václav Chvátal
Vertices 12
Edges 24
Radius 2
Diameter 2
Girth 4
Automorphisms 8 (D4)
Chromatic number 4
Chromatic index 4
Properties Regular
Hamiltonian
Triangle-free
Eulerian

In the mathematical field of graph theory, the Chvátal graph is an undirected graph with 12 vertices and 24 edges, discovered by Václav Chvátal (1970).

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 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, Branko Grünbaum (1970) 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 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.

An alternative conjecture of Bruce Reed (1998) 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

\chi(G)\le\left\lceil\frac{\Delta+\omega+1}{2}\right\rceil.

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 Fleischner & Sabidussi (2002) that it is NP-complete to determine whether a triangle-free Hamiltonian graph is 3-colorable.

The characteristic polynomial of the Chvátal graph is (x-4) (x-1)^4 x^2 (x+1) (x+3)^2 (x^2+x-4). The Tutte polynomial of the Chvátal graph has been computed by Björklund et al. (2008).

The independence number of this graph is 4.

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