Regular graphs with few longest cycles

TL;DR

This paper investigates the number of Hamiltonian cycles in regular graphs, establishing bounds, disproving conjectures, and exploring the relationship between connectivity, planarity, and cycle counts.

Contribution

It provides new results on the existence and bounds of Hamiltonian cycles in regular graphs with various connectivity and planarity conditions, and disproves a previous conjecture.

Findings

Existence of infinite families of 4-regular, 4-connected graphs with a fixed number of Hamiltonian cycles.

Disproof of Haythorpe's conjecture by constructing 5-regular graphs with exponentially many Hamiltonian cycles.

Identification of structural properties affecting the uniqueness of the longest cycle in 3-regular graphs.

Abstract

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput, and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every non-negative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} \cdot 3^{k+3}$ hamiltonian cycles. We prove that for every $d \ge 3$ there is an infinite family of hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components.