Few hamiltonian cycles in graphs with one or two vertex degrees

TL;DR

This paper disproves a conjecture about the minimum number of Hamiltonian cycles in certain graphs, extends existing methods using probabilistic techniques, and constructs infinite families of graphs with specific Hamiltonian properties.

Contribution

It fully disproves Haythorpe's conjecture, extends Thomassen's method with the Lovász Local Lemma, and constructs infinite non-regular graphs with exactly one Hamiltonian cycle.

Findings

Disproved Haythorpe's conjecture on Hamiltonian cycles.

Extended Thomassen's method using the Lovász Local Lemma.

Constructed infinite non-regular graphs with one Hamiltonian cycle.

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 $2k$.