Resonance Graph of Perfect Matchings

Abstract

Let G be a graph with perfect matchings and let C be a set of linearly independent even cycles of G of width at most 2. The resonance graph R(G, C) is a graph with the vertex set M a subset of M(G) such that two vertices Mi and Mj are adjacent if and only if the direct sum of Mi and Mj is E(c) for some cycle c in C. In this paper, we extend the results obtained by Tratnik and Ye to general graphs. Particulary, we show that the resonance graph of every graph with perfect matchings with respect to a set of linearly independent even cycles of width at most 2 is bipartite and each connected component of the resonance graph is an induced cubical graph.