Meredith graph

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Meredith graph
Meredith graph.svg
The Meredith graph
Named after G. H. Meredith
Vertices 70
Edges 140
Radius 7
Diameter 8
Girth 4
Automorphisms 38698352640
Chromatic number 3
Chromatic index 5
Book thickness 3
Queue number 2
Properties Eulerian
Table of graphs and parameters

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

Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.[4][5] However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.[6]

The characteristic polynomial of the Meredith graph is .

Gallery[edit]

References[edit]

  1. ^ Weisstein, Eric W. "Meredith graph". MathWorld. 
  2. ^ Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p. 470, 2007.
  3. ^ Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
  4. ^ Meredith, G. H. J. "Regular 4-Valent 4-Connected Nonhamiltonian Non-4-Edge-Colorable Graphs." J. Combin. Th. B 14, 55-60, 1973.
  5. ^ Bondy, J. A. and Murty, U. S. R. "Graph Theory with Applications". New York: North Holland, p. 239, 1976.
  6. ^ Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969.