Maximum matchings in regular graphs

TL;DR

This paper confirms a conjecture regarding maximum matchings in regular graphs, showing that for certain regular graphs, there exists a maximum matching with specific properties about unsaturated vertices sharing neighbors.

Contribution

It proves the conjecture for all k-regular simple graphs and for k-regular multigraphs with k ≤ 4, extending understanding of maximum matchings in regular graphs.

Findings

Confirmed the conjecture for all k-regular simple graphs.

Extended the result to k-regular multigraphs with k ≤ 4.

Established the existence of specific maximum matchings in these classes of graphs.

Abstract

It was conjectured by Mkrtchyan, Petrosyan, and Vardanyan that every graph $G$ with $\Delta(G)-\delta(G) \le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. In this note, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.