Characterization of Equimatchable Even-Regular Graphs

TL;DR

This paper characterizes connected equimatchable even-regular graphs, confirming the existence and uniqueness of such graphs with specific properties for even degrees greater than or equal to 6.

Contribution

It extends the classification of equimatchable regular graphs by establishing the existence and uniqueness of certain graphs with higher independence number and odd order for even degrees.

Findings

Existence of a unique connected equimatchable r-regular graph for even r ≥ 6

Such graphs have independence number at least 3 and odd order

Completes the classification for even-regular equimatchable graphs

Abstract

A graph is called equimatchable if all of its maximal matchings have the same size. Due to Eiben and Kotrb\v{c}\'{i}k,, any connected graph with odd order and independence number $\alpha(G)$ at most $2$ is equimatchable. Akbari et al. showed that for any odd number $r$, a connected equimatchable $r$-regular graph must be either the complete graph $K_{r+1}$ or the complete bipartite graph $K_{r,r}$. They also determined all connected equimatchable $4$-regular graphs and proved that for any even $r$, any connected equimatchable $r$-regular graph is either $K_{r,r}$ or factor-critical. In this paper, we confirm that for any even $r\ge 6$, there exists a unique connected equimatchable $r$-regular graph $G$ with $\alpha(G)\geq 3$ and odd order.