Few hamiltonian cycles in graphs with one or two vertex degrees
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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∈{ 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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