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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 WolframLanguage 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 ...

A complete graph is a graph in which each pair of graph vertices is connected by an edge. The complete graph with graph vertices is denoted and has (the triangular numbers) undirected edges, where is a binomial coefficient. In older literature, complete graphs are sometimes called universal graphs.The complete graph is also the complete n-partite graph .The complete graph on nodes is implemented in the Wolfram Language as CompleteGraph[n]. Precomputed properties are available using GraphData["Complete", n]. A graph may be tested to see if it is complete in the Wolfram Language using the function CompleteGraphQ[g].The complete graph on 0 nodes is a trivial graph known as the null graph, while the complete graph on 1 node is a trivial graph known as the singleton graph.In the 1890s, Walecki showed that complete graphs admit a Hamilton decomposition for odd , and decompositions into Hamiltonian cycles plus a perfect matching for..

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