On the Number of Cycles in a Graph

TL;DR

This paper investigates the extremal number of cycles in hamiltonian 3-connected cubic graphs, providing new bounds and proof techniques that could enhance understanding of cycle counts in such graphs.

Contribution

It establishes the smallest number of cycles in hamiltonian 3-connected cubic graphs and introduces a proof method to improve upper bounds on the maximum number of cycles.

Findings

Determined the minimum number of cycles in the specified graph class.

Proposed a proof technique to potentially improve upper bounds.

Contributed to the theoretical understanding of cycle counts in specialized graphs.

Abstract

There is a sizable literature on investigating the minimum and maximum numbers of cycles in a class of graphs. However, the answer is known only for special classes. This paper presents a result on the smallest number of cycles in hamiltonian 3-connected cubic graphs. Further, it describes a proof technique that could improve an upper bound of the largest number of cycles in a hamiltonian graph.