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Chvátal Graph


ChvatalGraph

Grünbaum conjectured that for every m>1, n>2, there exists an m-regular, m-chromatic graph of girth at least n. This result is trivial for n=2 or m=2,3, 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).

ChvatalGraphLCF

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 (-3)^2(1/2(-1-sqrt(17)))^1(-1)^10^21^4(1/2(-1+sqrt(17)))^14^1.


See also

Brinkmann Graph, Grünbaum Graphs, Quartic Graph

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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.

Cite this as:

Weisstein, Eric W. "Chvátal Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChvatalGraph.html

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