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Tutte Conjecture

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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.

NonhamiltonianBicubicGraphs

Known small counterexamples are summarized in the following table and illustrated above.

Vnamereference
50Georges graphGeorges (1989), Grünbaum (2006, 2009)
54Ellingham-Horton 54-graphEllingham and Horton (1983)
78Ellingham-Horton 78-graphEllingham (1981, 1982)
78Owens graphOwens (1983)
92Horton 92-graphHorton (1982)
96Horton 96-graphBondy and Murty (1976)

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

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, pp. 61 and 242, 1976.Bondy, 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 (n_3) 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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Tutte Conjecture

Cite this as:

Weisstein, Eric W. "Tutte Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TutteConjecture.html

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