Forcing hexagons in hexagonal systems

摘要

We introduce the concept of a forcing hexagon in a hexagonal system H, which is a hexagon h in H such that the subgraph of H obtained by deleting all vertices of h together with their incident edges has exactly one perfect matching. We show that any hexagonal system with a forcing hexagon is a normal hexagonal system. We further prove that every hexagon of a hexagonal system H is forcing if and only if H is a linear hexagonal chain, and that any other hexagonal system has at most two forcing hexagons. Using the tool of Z-transformation graphs developed by F. Zhang et al, we prove the co-existence property of forcing hexagons and forcing edges, and we obtain the structural characterizations for the hexagonal systems with a given number of forcing hexagons. Miscellaneous related results are presented. We also post a question for further investigation.