Equimatchable Regular Graphs

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,K- 3. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r-regular graph, then G is an element of{Kr+1,K-r,K-r}. 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 K-r,K-r. (C) 2017 Wiley Periodicals, Inc.