Triangle-free equimatchable graphs

A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. provided a characterization of equimatchable graphs with girth at least 5. In this paper, we extend this result by providing a complete structural characterization of equimatchable graphs with girth at least 4, that is, equimatchable graphs with no triangle, by identifying the equimatchable triangle-free graph families. Our characterization also extends the result given by Akbari et al., which proves that the only connected triangle-free equimatchable r-regular graphs are C 5, C 7, and K r , r, where r is a positive integer. Given a nonbipartite graph, our characterization implies a linear time recognition algorithm for triangle-free equimatchable graphs.