# Ellingham–Horton graph

Ellingham–Horton graphs
The Ellingham–Horton 54-graph.
Named after Joseph Horton and Mark Ellingham
Vertices 54 (54-graph)
78 (78-graph)
Edges 81 (54-graph)
117 (78-graph)
7 (78-graph)
Diameter 10 (54-graph)
13 (78-graph)
Girth 6 (both)
Automorphisms 32 (54-graph)
16 (78-graph)
Chromatic number 2 (both)
Chromatic index 3 (both)
Properties Cubic (both)
Bipartite (both)
Regular (both)

In the mathematical field of graph theory, the Ellingham–Horton graphs are two 3-regular graphs on 54 and 78 vertices : the Ellingham–Horton 54-graph and the Ellingham–Horton 78-graph.[1] They are named after Joseph D. Horton and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian.[2]

The first counterexample to the Tutte conjecture was the Horton graph, published by Bondy & Murty (1976).[3] After the Horton graph, a number of smaller counterexamples to the Tutte conjecture were found. Among them are a 92-vertex graph by Horton (1982),[4] a 78-vertex graph by Owens (1983),[5] and the two Ellingham–Horton graphs.

The first Ellingham–Horton graph was published by Ellingham (1981) and is of order 78.[6] At that time it was the smallest known counterexample to the Tutte conjecture. The second Ellingham–Horton graph was published by Ellingham & Horton (1983) and is of order 54.[7] In 1989, Georges' graph, the smallest currently-known Non-Hamiltonian 3-connected cubic bipartite graph was discovered, containing 50 vertices.[8]

## References

1. ^
2. ^ Tutte, W. T. (1971), "On the 2-factors of bicubic graphs", Discrete Mathematics 1 (2): 203–208, doi:10.1016/0012-365X(71)90027-6.
3. ^ Bondy, J. A.; Murty, U. S. R. (1976), Graph Theory with Applications, New York: North Holland, p. 240, ISBN 0-444-19451-7
4. ^ Horton, J. D. (1982), "On two-factors of bipartite regular graphs", Discrete Mathematics 41 (1): 35–41, doi:10.1016/0012-365X(82)90079-6.
5. ^ Owens, P. J. (1983), "Bipartite cubic graphs and a shortness exponent", Discrete Mathematics 44 (3): 327–330, doi:10.1016/0012-365X(83)90201-7.
6. ^ Ellingham, M. N. (1981), Non-Hamiltonian 3-connected cubic partite graphs, Research Report 28, Melbourne: Dept. of Math., Univ. Melbourne.
7. ^ Ellingham, M. N.; Horton, J. D. (1983), "Non-Hamiltonian 3-connected cubic bipartite graphs", Journal of Combinatorial Theory, Series B 34 (3): 350–353, doi:10.1016/0095-8956(83)90046-1.
8. ^ Georges, J. P. (1989), "Non-hamiltonian bicubic graphs", Journal of Combinatorial Theory, Series B 46 (1): 121–124, doi:10.1016/0095-8956(89)90012-9.