Connected cubic graphs with the maximum number of perfect matchings

TL;DR

This paper establishes an upper bound on the number of perfect matchings in connected cubic graphs, characterizes the extremal graph, relates perfect matchings to edge colorings, and improves bounds on cycles.

Contribution

It proves a maximum bound on perfect matchings in connected cubic graphs, characterizes the extremal graph, and links perfect matchings to edge colorings.

Findings

Maximum number of perfect matchings is at most 4 times a Fibonacci number.

The extremal graph achieving this maximum is uniquely characterized.

Provides an improved lower bound on the number of cycles in cubic graphs.

Abstract

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well. In addition, it is shown that the number of perfect matchings in any cubic graph $G$ equals the expected value of a random variable defined on all $2$-colorings of edges of $G$. Finally, an improved lower bound on the maximum number of cycles in a cubic graph is provided.