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.
See also
Bicubic Graph, Bicubic Nonhamiltonian Graph, Cubic Graph, Ellingham-Horton Graphs, Georges Graph, Horton Graphs, Nonhamiltonian Graph, Tait's Hamiltonian Graph Conjecture
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References
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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. B 34, 350-353, 1983.Georges,
J. P. "Non-Hamiltonian Bicubic Graphs." J. Combin. Th. B 46,
121-124, 1989.Gropp, H. "Configurations and the Tutte Conjecture."
Ars. Combin. A 29, 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.
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Cite this as:
Weisstein, Eric W. "Tutte Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TutteConjecture.html