Chvátal graph
| Chvátal graph | |
|---|---|
 |
|
| 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.
...
Wikipedia