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# Few hamiltonian cycles in graphs with one or two vertex degrees

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## Abstract

We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe's computational enumerative results from [Experim. Math. 27 (2018) 426-430]. Thereafter, we use the Lov\'asz Local Lemma to extend Thomassen's independent dominating set method. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen. Motivated by an observation of Aldred and Thomassen, we prove that for every $\kappa \in \{ 2, 3 \}$ and any positive integer $k$, there are infinitely many non-regular graphs of connectivity $\kappa$ containing exactly one hamiltonian cycle and in which every vertex has degree $3$ or $2k$.

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A note on independent dominating sets and second hamiltonian cycles
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M. Ghandehari and H. Hatami. A note on independent dominating sets and second hamiltonian cycles. Manuscript.
Code and certificates for the paper "Few hamiltonian cycles in graphs with one or two vertex degrees
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• J Jooken
• O.-H S Lo
• B Seamone
• C T Zamfirescu
J. Goedgebeur, J. Jooken, O.-H. S. Lo, B. Seamone, and C. T. Zamfirescu. Code and certificates for the paper "Few hamiltonian cycles in graphs with one or two vertex degrees" (2022). See: https://github.com/JorikJooken/hamiltonian_cycles
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• J Goedgebeur
• J Renders
• G Wiener
• C T Zamfirescu
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G. Royle. The smallest uniquely hamiltonian graph with minimum degree at least 3 (2017). https://mathoverflow.net/questions/255784/ what-is-the-smallest-uniquely-hamiltonian-graph-with-minimum-degree-at-least-3/