Tutte (1971/72) conjectured that there are no 3-connected nonhamiltonian bicubic graphs. However, a counterexample was found by J. D. Horton
in 1976 (Gropp 1990), and several smaller counterexamples are now known.
Known small counterexamples are summarized in the following table and illustrated above.
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J. A. and Murty, U. S. R. Graph
Theory. Berlin: Springer-Verlag, pp. 487-488, 2008.Ellingham,
M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs." Research
Report No. 28, Dept. of Math., Univ. Melbourne, Melbourne, 1981.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, 1982.Ellingham,
M. N. and Horton, J. D. "Non-Hamiltonian 3-Connected Cubic Bipartite
Graphs." J. Combin. Th. Ser. B34, 350-353, 1983.Georges,
J. P. "Non-Hamiltonian Bicubic Graphs." J. Combin. Th. B46,
121-124, 1989.Gropp, H. "Configurations and the Tutte Conjecture."
Ars. Combin. A29, 171-177, 1990.Grünbaum, B. "3-Connected
Configurations
with No Hamiltonian Circuit." Bull. Inst. Combin. Appl.46, 15-26,
2006.Grünbaum, B. Configurations
of Points and Lines. Providence, RI: Amer. Math. Soc., p. 311, 2009.Horton,
J. D. "On Two-Factors of Bipartite Regular Graphs." Disc. Math.41,
35-41, 1982.Owens, P. J. "Bipartite Cubic Graphs and a Shortness
Exponent." Disc. Math.44, 327-330, 1983.Tutte, W. T.
"On the 2-Factors of Bicubic Graphs." Disc. Math.1, 203-208,
1971/72.