Equimatchable Regular Graphs

Abstract

Abstract A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3‐regular graphs are K 4 and K 3, 3 . We extend this result by showing that for an odd positive integer r , if G is a connected equimatchable r ‐regular graph, then . Also it is proved that for an even r , a connected triangle‐free equimatchable r ‐regular graph is isomorphic to one of the graphs C 5 , C 7 , and .