On the Number of Matchings in Regular Graphs

TL;DR

This paper investigates the extremal number of matchings in regular graphs, characterizes optimal structures, and provides proofs for specific cases, advancing understanding of matching counts in graph theory.

Contribution

It characterizes graphs with extremal numbers of matchings for certain degree sequences and proves conjectures for specific regular bipartite graphs and matching sizes.

Findings

Identifies graphs maximizing and minimizing m-matchings for given degree sequences.

Calculates expected number of m-matchings in r-regular bipartite graphs.

Proves conjectured bounds for 2-regular bipartite graphs and small m-matchings.

Abstract

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for 2-regular bipartite graphs and for $m$-matchings with $m\le 4$.