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On uniquely Hamiltonian claw-free and triangle-free graphs. (English) Zbl 1311.05104
Summary: A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to R. C. Entringer and H. Swart [J. Comb. Theory, Ser. B 29, 303–309 (1980; Zbl 0387.05017)] can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.

05C45 Eulerian and Hamiltonian graphs
05C38 Paths and cycles
Full Text: DOI
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