Grünbaum Graphs

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 4. It is Hamiltonian and nonplanar.

Two other graphs associated with Grünbaum are the graphs on 121 and 124 vertices illustrated above (Grünbaum 1970a, Zamfirescu 1976). They are implemented in the Wolfram Language as GraphData["GruenbaumGraph121"] and GraphData["GruenbaumGraph124"]. The 124-vertex graph is nonhamiltonian and is therefore a counterexample to the Tait's Hamiltonian graph conjecture.

REFERENCES

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 241-242, 1976.

Grünbaum, B. "Polytopes, Graphs, and Complexes." Bull. Amer. Math. Soc. 76, 1131-1201, 1970a.

Grünbaum, B. "A Problem in Graph Coloring." Amer. Math. Monthly 77, 1088-1092, 1970b.Zamfirescu, T. "On Longest Paths and Circuits in Graphs." Math. Scand. 38, 211-239, 1976.