Maximally resonant polygonal systems

A benzenoid system GG is kk-resonant if any set FF of no more than kk disjoint hexagons is a resonant pattern, i.e, G−FG−F has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are kk-resonant for all k≥1k≥1. Recently, the equivalence of 3-resonance and maximal resonance has been shown to be valid also for coronoid systems, carbon nanotubes, polyhexes in tori and Klein bottles, and fullerene graphs. So our main problem is to investigate the extent of graphs possessing this interesting property. In this paper, by replacing the above hexagons with even faces, we define kk-resonance of graphs in surfaces, possibly with boundary, in a unified way. Some exceptions exist. For plane polygonal systems tessellated with polygons of even size at least six such that all inner vertices have the same degree three and the others have degree two or three, we show that such 3-resonant polygonal systems are indeed maximally resonant. They can be constructed by gluing and lapping operations on three types of basic graphs.