Resonance Graphs on Perfect Matchings of Graphs on Surfaces

Let G be a graph embedded in a surface and let be the set of all faces of G . For a given subset ℱ of even faces (faces bounded by an even cycle), 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 M_1 and M_2 are adjacent if and only if the symmetric difference M_1⊕ M_2 is a cycle bounding some face in ℱ . It has been shown that if G is a plane elementary 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. However, this result does not hold in general for plane graphs G . The structure properties of resonance graphs in general remain unknown. In this paper, we show that every connected component of the resonance graph of a graph G on a surface with respect to a given even-face set can always be embedded into a hypercube as an induced subgraph. Further, 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) .