Resonance Graphs and Perfect Matchings of Graphs on Surfaces

Let G be a graph embedded in a surface and let F be a set of even faces of G (faces bounded by a cycle of even length). The resonance graph of G with respect to F, denoted by R(G;F), is a graph such that its vertex set is the set of all perfect matchings of G and two vertices M1 and M2 are adjacent to each other if and only if the symmetric difference M1 ⊕M2 is a cycle bounding some face in F. It has been shown that if G is a matching-covered plane bipartite graph, the resonance graph of G with respect to the set of all inner faces is isomorphic to the covering graph of a distributive lattice. It is evident that the resonance graph of a plane graph G with respect to an even-face set F may not be the covering graph of a distributive lattice. In this paper, we show the resonance graph of a graph G on a surface with respect to a given even-face set F can always be embedded into a hypercube as an induced subgraph. Furthermore, we show that the Clar covering polynomial of G with respect to F is equal to the cube polynomial of the resonance graph R(G;F), which generalizes previous results on some subfamilies of plane graphs.