Abstract
AbstractA 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 ‐regular graphs are , , and , where is a positive integer. Given a nonbipartite graph, our characterization implies a linear time recognition algorithm for triangle‐free equimatchable graphs.
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