# Meredith graph

Meredith graph
The Meredith graph
Named after G. H. Meredith
Vertices 70
Edges 140
Diameter 8
Girth 4
Automorphisms 38698352640
Chromatic number 3
Chromatic index 5
Properties Eulerian

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.[1]

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.[2]

Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.[3][4] However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.[5]

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

## References

1. ^
2. ^ Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p. 470, 2007.
3. ^ Meredith, G. H. J. "Regular 4-Valent 4-Connected Nonhamiltonian Non-4-Edge-Colorable Graphs." J. Combin. Th. B 14, 55-60, 1973.
4. ^ Bondy, J. A. and Murty, U. S. R. "Graph Theory with Applications". New York: North Holland, p. 239, 1976.
5. ^ Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969.