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.Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p. 470, 2007. It has book thickness 3 and queue number 2.Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.Bondy, J. A. and Murty, U. S. R. "Graph Theory with Applications". New York: North Holland, p. 239, 1976. However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969.
The characteristic polynomial of the Meredith graph is .
Gallery
References