Few hamiltonian cycles in graphs with one or two vertex degrees

Inspired by Sheehan’s conjecture that no 4 4 -regular graph contains exactly one hamiltonian cycle, we prove results on hamiltonian cycles in regular graphs and nearly regular graphs. 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 [Exp. Math. 27 (2018), no. 4, 426–430]. Thereafter, we use the Lovász Local Lemma to extend Thomassen’s independent dominating set method. This extension allows us to find a second hamiltonian cycle that inherits linearly many edges from the first hamiltonian cycle. 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 by showing that for k ∈ { 5 , 6 } k \in \{5, 6\} there exist infinitely many k k -regular hamiltonian graphs having no independent dominating set with respect to a prescribed hamiltonian cycle. Motivated by an observation of Aldred and Thomassen, we prove that for every κ ∈ { 2 , 3 } \kappa \in \{ 2, 3 \} and any positive integer k k , there are infinitely many non-regular graphs of connectivity κ \kappa containing exactly one hamiltonian cycle and in which every vertex has degree 3 3 or 2 k 2k .