Geodesic
Equimatchable bipartite graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 77-94

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A graph is called equimatchable if all of its maximal matchings have the same size. Lesk et al. [Equi-matchable graphs, Graph Theory and Combinatorics (Academic Press, London, 1984) 239–254] has provided a characterization of equimatchable bipartite graphs. Motivated by the fact that this characterization is not structural, Frendrup et al. [A note on equimatchable graphs, Australas. J. Combin. 46 (2010) 185–190] has 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 paper, we extend the characterization of Frendrup by eliminating the girth condition. For an equimatchable graph, an edge is said to be a critical-edge if the graph obtained by the removal of this edge is not equimatchable. An equimatchable graph is called edge-critical, denoted by ECE, if every edge is critical. Noting that each ECE-graph can be obtained from some equimatchable graph by recursively removing non-critical edges, each equimatchable graph can also be constructed from some ECE-graph by joining some non-adjacent vertices. Our study reduces the characterization of equimatchable bipartite graphs to the characterization of bipartite ECE-graphs.
Keywords: equimatchable, edge-critical, bipartite graphs 
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Büyükçolak, Yasemin; Gözüpek, Didem; Özkan, Sibel. Equimatchable bipartite graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 77-94. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a4/