There are (at least) two graphs associated with Ellingham and Horton. These graphs on 54 and 78 nodes respectively, illustrated above, are examples of 3-connected bicubic nonhamiltonian graphs, and therefore
provide counterexamples to the Tutte conjecture.
Ellingham, M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs." Research Report No. 28, Dept. of Math., Univ. Melbourne,
Melbourne, 1981.Ellingham, M. N. Cycles in 3-Connected Cubics
Graphs. M.Sc. thesis. Melbourne, Australia: University of Melbourne, June 1982a.Ellingham,
M. N. "Constructing Certain Cubic Graphs." In Combinatorial Mathematics,
IX: Proceedings of the Ninth Australian Conference held at the University of Queensland,
Brisbane, August 24-28, 1981 (Ed. E. J. Billington, S. Oates-Williams,
and A. P. Street). Berlin: Springer-Verlag, pp. 252-274, 1982b.Ellingham,
M. N. and Horton, J. D. "Non-Hamiltonian 3-Connected Cubic Bipartite
Graphs." J. Combin. Th. Ser. B34, 350-353, 1983.
