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 
.
