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arXiv · 2211.08105

Few hamiltonian cycles in graphs with one or two vertex degrees

Jan Goedgebeur·Jorik Jooken·On-Hei Solomon Lo·Ben Seamone·Carol T. Zamfirescu
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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 Lovász 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 $κ\in \{ 2, 3 \}$ and any positive integer $k$, there are infinitely many non-regular graphs of connectivity $κ$ containing exactly one hamiltonian cycle and in which every vertex has degree $3$ or $2k$.

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Jan Goedgebeur, Jorik Jooken, On-Hei Solomon Lo, Ben Seamone, Carol T. Zamfirescu. 2022-11-15. Few hamiltonian cycles in graphs with one or two vertex degrees. https://arxiv.org/abs/2211.08105

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