| Ellingham–Horton graphs | |
| Namesake: | Joseph Horton and Mark Ellingham |
| Vertices: | 54 (54-graph) 78 (78-graph) |
| Edges: | 81 (54-graph) 117 (78-graph) |
| Automorphisms: | 32 (54-graph) 16 (78-graph) |
| Girth: | 6 (both) |
| Diameter: | 10 (54-graph) 13 (78-graph) |
| Radius: | 9 (54-graph) 7 (78-graph) |
| Chromatic Number: | 2 (both) |
| Chromatic Index: | 3 (both) |
| Book Thickness: | 3 (both) |
| Queue Number: | 2 (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. 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.[1] The book thickness of the Ellingham-Horton 54-graph and the Ellingham-Horton 78-graph is 3 and the queue numbers 2.[2]
The first counterexample to the Tutte conjecture was the Horton graph, published by . After the Horton graph, a number of smaller counterexamples to the Tutte conjecture were found. Among them are a 92-vertex graph by,[3] a 78-vertex graph by,[4] and the two Ellingham–Horton graphs.
The first Ellingham–Horton graph was published by and is of order 78.[5] At that time it was the smallest known counterexample to the Tutte conjecture. The second Ellingham–Horton graph was published by and is of order 54.[6] In 1989, Georges' graph, the smallest currently-known Non-Hamiltonian 3-connected cubic bipartite graph was discovered, containing 50 vertices.[7]