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Independent dominating sets and Hamiltonian cycles. (English) Zbl 1112.05077
Summary: A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note we prove that there are no \(r\)-regular uniquely Hamiltonian graphs when \(r > 22\). This improves upon earlier results of C. Thomassen [J. Comb. Theory, Ser. B 72, 104–109 (1998; Zbl 0895.05038)].

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C45 Eulerian and Hamiltonian graphs
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[1] Bondy, J Combin Theory Ser B 74 pp 265– (1998)
[2] and , Combinatorial enumeration, Wiley-Interscience series in discrete mathematics. John Wiley & Sons, Inc., New York, 1983.
[3] and , A note on independent dominating sets and second hamiltonian cycles (Personal communication).
[4] and , Problems and results on 3-chromatic hypergraphs and some related questions, In: Infinite and finite sets, et al., eds., North-Holland, Amsterdam, 1975, 609–628.
[5] The multiplicity of Hamiltonian circuits in a graph, In: Recent advances in graph theory, ed., Academia, Prague, 1975, 447–480.
[6] Tutte, J London Math Soc 21 pp 98– (1946)
[7] Thomason, Ann Discrete Math 3 pp 3– (1978)
[8] Thomassen, J Combin Theory Ser B 72 pp 104– (1998)
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