Meredith graph

In the mathematical field of graph theory, the Meredith graph is a 4-regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973.

The Meredith graph is 4-vertex-connected and 4-edge-connected, has chromatic number 3, chromatic index 5, radius 7, diameter 8, girth 4 and is non-Hamiltonian. It has book thickness 3 and queue number 2.

Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian. However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.

The characteristic polynomial of the Meredith graph is $$(x-4) (x-1)^{10} x^{21} (x+1)^{11} (x+3) (x^2-13) (x^6-26 x^4+3 x^3+169 x^2-39 x-45)^4$$.