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 Kotrbcik, any connected graph with odd order and independence number α(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 Kᵣ₊₁ or the complete bipartite graph Kr,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 Kr,r or factor-critical. In this paper, we confirm that for any even r≥ 6, there exists a unique connected equimatchable r-regular graph G with α(G)≥ 3 and odd order.