Regular coronoids and 4-tilings

Aleksander Vesel
University of Maribor, IMFM

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Minisymposium: CHEMICAL GRAPH THEORY

Content: A benzenoid graph is a finite connected plane graph with no cut-vertices in which every interior region is bounded by a regular hexagon of a side length one. A coronoid $G$ is a connected subgraph of a benzenoid graph such that every edge lies in a hexagon of $G$ and $G$ contains at least one non-hexagon interior face (called corona hole), which should have a size of at least two hexagons. A polyhex is either a benzenoid or a coronoid. Regular coronoids can be generated from a single hexagon by a series of normal additions plus corona condensations of modes $L_2$ or $A_2$. The vertices of the inner dual of a polyhex $G$, denoted by $I(G)$, are the centers of all hexagons of $G$, two vertices being adjacent if and only if the corresponding hexagons share an edge in $G$. Let $S$ denote a subset of internal edges of $E(I(G))$. If $I_S(G) := I(G) \setminus S$ is the graph where for every triple of pairwise adjacent hexagons $h_1, h_2, h_3$ of $V(I(G))$ there exists a hexagon $h_4$ of $V(I(G))$ such that $h_1, h_2, h_3, h_4$ induce a 4-cycle of $I_S(G)$, then $S$ is a 4-tiling of $G$. We show that a coronoid $G$ admits a 4-tiling if and only if $G$ is regular. Moreover, we 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 $A_2$. This confirms Conjecture 6 stated in [H.~Zhang, Regular coronoids and ear decompositions of plane elementary bipartite graphs, {\em CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory } (2007) 259--271].

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