Meredith graph explained

Meredith graph
Namesake:	G. H. Meredith
Vertices:	70
Edges:	140
Girth:	4
Diameter:	8
Radius:	7
Automorphisms:	38698352640
Chromatic Number:	3
Chromatic Index:	5
Properties:	Eulerian
Book Thickness:	3
Queue Number:	2

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.^[1] It has book thickness 3 and queue number 2.^[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

(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} .

Notes and References

1. Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p. 470, 2007.
2. Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
3. Meredith . G. H. J. . 10.1016/s0095-8956(73)80006-1 . Journal of Combinatorial Theory, Series B . 311503 . 55–60 . Regular -valent -connected nonHamiltonian non--edge-colorable graphs . 14 . 1973. free .
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.