On the Minimum Number of Hamiltonian Cycles in Regular Graphs

TL;DR

This paper introduces specific constructions of k-regular graphs and explicitly calculates their Hamiltonian cycles, providing bounds and insights into the minimum number of such cycles in regular graphs.

Contribution

It presents new graph constructions and formulas for Hamiltonian cycles, offering tight bounds and comparative analysis for regular graphs with k >= 3.

Findings

Explicit formulas for Hamiltonian cycles in constructed graphs

Empirical evidence for tight upper bounds when k >= 5

A new construction for 4-regular graphs with better cycle counts

Abstract

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and k, and empirical evidence is provided that suggests that this function gives a tight upper bound on the minimum number of Hamiltonian cycles in k-regular graphs on n vertices for k >= 5 and n >= k + 3. An additional graph construction for 4-regular graphs is described for which the number of Hamiltonian cycles is superior to the above function in the case when k = 4 and n >= 11.