Regular coronoids and 4-tilings

A benzenoid is a finite connected plane graph with no cut vertices in which every interior region is bounded by a regular hexagon of side length one. A coronoid G is a connected subgraph of a benzenoid such that every edge lies in a hexagon of G and G contains at least one non-hexagon interior face, which should have a size of at least two hexagons. A polyhex is either a benzenoid or a coronoid. A coronoid is said to be regular if it can be generated from a single hexagon by a series of specific additions of hexagons. The vertices of the inner dual I(G) of a polyhex G are the centers of all hexagons of G, two vertices being adjacent if and only if the corresponding hexagons share an edge in G. The graph IS(G) is obtained from I(G) by removing a set of internal edges S. A polyhex G admits a 4-tiling, if every triple of pairwise adjacent hexagons of G belongs to a 4-cycle face of IS(G). We show that a coronoid G admits a 4-tiling if and only if G is regular. This fact enables us to prove that a coronoid G is regular if and only if G can be generated from a single hexagon by a series of normal additions plus corona condensations of mode A2. This result confirms Conjecture 6 stated in [9].