2-resonance of plane bipartite graphs and its applications to boron–nitrogen fullerenes

A set HH of disjoint faces of a plane bipartite graph GG is a resonant pattern if GG has a perfect matching MM such that the boundary of each face in HH is an MM-alternating cycle. An elementary result was obtained [Discrete Appl. Math. 105 (2000) 291–311]: a plane bipartite graph is 1-extendable if and only if every face forms a resonant pattern. In this paper we show that for a 2-extendable plane bipartite graph, any pair of disjoint faces form a resonant pattern, and the converse does not necessarily hold. As an application, we show that all boron–nitrogen (B–N) fullerene graphs are 2-resonant, and construct all the 3-resonant B–N fullerene graphs, which are all kk-resonant for any positive integer kk. Here a B–N fullerene graph is a plane cubic graph with only square and hexagonal faces, and a B–N fullerene graph is kk-resonant if any i(0⩽i⩽k) disjoint faces form a resonant pattern. Finally, the cell polynomials of 3-resonant B–N fullerene graphs are computed.