Maximal resonance of cubic bipartite polyhedral graphs

Let be a set of disjoint faces of a cubic bipartite polyhedral graph G . If G has a perfect matching M such that the boundary of each face of is an M -alternating cycle (or in other words, has a perfect matching), then is called a resonant pattern of G . Furthermore, G is k -resonant if every disjoint faces of G form a resonant pattern. In particular, G is called maximally resonant if G is k -resonant for all integers . In this paper, all the cubic bipartite polyhedral graphs, which are maximally resonant, are characterized. As a corollary, it is shown that if a cubic bipartite polyhedral graph is 3-resonant then it must be maximally resonant. However, 2-resonant ones need not to be maximally resonant.