Triangle-Free Equimatchable Graphs

07/25/2018

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by   Yasemin Büyükçolak, et al.

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A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. [8] 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, i.e., equimatchable graphs with no triangle, by identifying the equimatchable triangle-free graph families. Our characterization also extends the result given by Akbari et al. in [1], 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 non-bipartite graph, our characterization implies a linear time recognition algorithm for triangle-free equimatchable graphs.