Chvátal Graph

Grünbaum conjectured that for every , , there exists an -regular, -chromatic graph of girth at least . This result is trivial for  or , but only a small number of other such graphs are known, including the 12-node Chvátal graph, 21-node Brinkmann graph, and 25-node Grünbaum graph. The Chvátal graph is illustrated above in a couple embeddings (e.g., Bondy; Knuth 2008, p. 39).

It has 370 distinct (directed) Hamiltonian cycles, giving a unique generalized LCF notation of order 4 (illustrated above), two of order 6 (illustrated above), and 43 of order 1.

The Chvátal graph is implemented in the Wolfram Language as GraphData["ChvatalGraph"].

The Chvátal graph is a quartic graph on 12 nodes and 24 edges. It has chromatic number 4, and girth 4. The Chvátal graph has graph spectrum .

REFERENCES

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 241, 1976.

Grünbaum, B. "A Problem in Graph Coloring." Amer. Math. Monthly 77, 1088-1092, 1970.

Knuth, D. E. The Art of Computer Programming, Volume 4, Fascicle 0: Introduction to Combinatorial Functions and Boolean Functions.. Upper Saddle River, NJ: Addison-Wesley, 2008.