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Flip-Flops in Hypohamiltonian Graphs

Published online by Cambridge University Press:  20 November 2018

V. Chvátal*
Affiliation:
Centre De Recherches Mathématiques,Montréal Québec
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Throughout this note, we adopt the graph-theoretical terminology and notation of Harary [3]. A graph G is hypohamiltonianif G is not hamiltonian but the deletion of any point u from G results in a hamiltonian graph G-u. Gaudin, Herz, and Rossi [2] proved that the smallest hypohamiltonian graph is the Petersen graph. Using a computer for a systematic search, Herz, Duby, and Vigué [4] found that there is no hypohamiltonian graph with 11 or 12 points. However, they found one with 13 and one with 15 points. Sousselier [4] and Lindgren [5] constructed independently the same sequence of hypohamiltonian graphs with 6k+10 points. Moreover, Sousselier found a cubic hypohamiltonian graph with 18 points. This graph and the Petersen graph were the only examples of cubic hypohamiltonian graphs until Bondy [1] constructed an infinite sequence of cubic hypohamiltonian graphs with 12k+10 points. Bondy also proved that the Coxeter graph [6], which is cubic with 28 points, is hypohamiltonian.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Bondy, J. A., Variations on the Hamiltonian theme, Canad. Math. Bull. (1) 15 (1972), 5762.Google Scholar
2. Gaudin, T., Herz, J.-C. and Rossi, P., Solution du Problème no. 29, Rev. Française Informat Recherche Opérationnelle,8 (1964), 214218.Google Scholar
3. Harary, F., Graph theory, Addison-Wesley, Reading, Mass., 1969.Google Scholar
4. Herz, J.-C., Duby, J.-J. and Vigué, F., Recherche systématique des graphes Hypohamiltoniens, in Theory of Graphs (edited by Rosenstiehl, P.), International Symposium, Rome (1966), 153159.Google Scholar
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