Edge-Critical Equimatchable Bipartite Graphs

Abstract

A graph is called equimatchable if all of its maximal matchings have the same size. Lesk et al. [6] provided a characterization of equimatchable bipartite graphs. Since this characterization is not structural, Frendrup et al. [4] also provided a structural characterization for equimatchable graphs with girth at least five; in particular, a characterization for equimatchable bipartite graphs with girth at least six. In this work, we extend the partial characterization of Frendrup et al. [4] to equimatchable bipartite graphs without any restriction on girth. For an equimatchable graph, an edge is said to be critical-edge if the graph obtained by removal of this edge is not equimatchable. An equimatchable graph is called edge-critical if every edge is critical. Reducing the characterization of equimatchable bipartite graphs to the characterization of edge-critical equimatchable bipartite graphs, we give two characterizations of edge-critical equimatchable bipartite graphs.