Resonance Graphs and Perfect Matchings of Graphs on Surfaces

TL;DR

This paper studies resonance graphs of graphs embedded on surfaces, showing they can always be embedded into a hypercube and relating their structure to the Clar covering polynomial, generalizing known results.

Contribution

It proves that resonance graphs of surface-embedded graphs can be embedded into hypercubes and links their cube polynomial to the Clar covering polynomial, extending previous planar graph results.

Findings

Resonance graphs can be embedded into hypercubes as induced subgraphs.

The cube polynomial of the resonance graph equals the Clar covering polynomial.

Generalizes known planar graph results to graphs on arbitrary surfaces.

Abstract

Let $G$ be a graph embedded in a surface and let $\mathcal F$ be a set of even faces of $G$ (faces bounded by a cycle of even length). The resonance graph of $G$ with respect to $\mathcal F$, 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 to each other 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 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 $\mathcal F$ 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 $\mathcal F$ can always be embedded into a hypercube as an induced subgraph. Furthermore, we show that the Clar covering polynomial of $G$ with respect to $\mathcal F$ is equal to the cube polynomial of the resonance graph $R(G;\mathcal F)$, which generalizes previous results on some subfamilies of plane graphs.