Characterization of Equimatchable Even-Regular Graphs

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.