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 work, 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 C5, C7 and Kr,r, where r is a positive integer. Given a non-bipartite graph, our characterization implies a linear time recognition algorithm for triangle-free equimatchable graphs.

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