# Regular coronoids and 4-tilings

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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].