The Meredith graph is a quartic graph on 70 nodes and 140 edges that is a counterexample to the conjecture that every 4-regular 4-connected graph is Hamiltonian.It is implemented in the Wolfram Languageas GraphData["MeredithGraph"].The Meredith graph has chromatic number 3 andedge chromatic number 5.The plots above show the adjacency, incidence,and distance matrices of the graph.
Grünbaum conjectured that for every , , there exists an -regular, -chromatic graph of girth at least . This result is trivial for and , but only a small number of other such graphs are known, including the Grünbaum graph, illustrated above, Brinkmann graph, and Chvátal graph.The Grünbaum graph can be constructed from the dodecahedral graph by adding an additional ring of five vertices around the perimeter and cyclically connecting each new vertex to three others as shown above (left figure). A more symmetrical embedding is shown in the center figure above, and an LCF notation-based embedding is shown in the right figure. This graph is implemented in the Wolfram Language as GraphData["GruenbaumGraph25"].The Grünbaum graph has 25 vertices and 50 edges. It is a quartic graph with chromatic number 4, and therefore has . It has girth .It has diameter 4, graph radius 3, edge connectivity 4, and vertex connectivity..