Resonance Graphs on Perfect Matchings of Graphs on Surfaces

Abstract

Let G be a graph embedded in a surface and let be the set of all faces of G. For a given subset $$\mathcal F$$ of even faces (faces bounded by an even cycle), the resonance graph of G with respect to $${\mathcal {F}}$$ , denoted by $$R(G;\mathcal F)$$ , 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 $$M_1\oplus M_2$$ is a cycle bounding some face in $${\mathcal {F}}$$ . 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 $${\mathcal {F}}$$ is equal to the cube polynomial of the resonance graph $$R(G;{\mathcal {F}})$$ .