Abstract
Let G be a plane bipartite graph which admits a perfect matching and with distinguished faces called holes. Let M G denote the perfect matchings graph: its vertices are the perfect matchings of G, two of them being joined by an edge, if and only if they differ only on an alternating cycle bounding a face which is not a hole. We solve the following problem: Find a criterion for two perfect matchings of G to belong to the same connected component of M G , and in particular determine in which case M G is connected. The motivation of this work is a result on tilings of Saldanha et al. (Comput. Geom. 14 (1995) 207).
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