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Flip-Flops in Hypohamiltonian Graphs

  • V. Chvátal (a1)
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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.

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References
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1. Bondy, J. A., Variations on the Hamiltonian theme, Canad. Math. Bull. (1) 15 (1972), 5762.
2. Gaudin, T., Herz, J.-C. and Rossi, P., Solution du Problème no. 29, Rev. Française Informat Recherche Opérationnelle,8 (1964), 214218.
3. Harary, F., Graph theory, Addison-Wesley, Reading, Mass., 1969.
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.
5. Lindgren, W. F., An infinite class of Hypohamiltonian graphs, Amer. Math. Monthly, 74 (1967), 10871088.
6. Tutte, W. T., A non-Hamiltonian graph, Canad. Math. Bull. 3 (1960), 15.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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