Extremal fullerene graphs with the maximum Clar number

Details Extremal fullerene graphs with the maximum Clar number Extremal fullerene graphs with the maximum Clar number

2008-01-11

arxiv.org/abs/0801.1788v2

Discrete Appl. Math. 157 (2009) 3152-3173

Mathematics

Combinatorics

35 pages, 43 figures

Scientific paper

10.1016/j.dam.2009.06.007

A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let $F_n$ be a fullerene graph with $n$ vertices. A set $\mathcal H$ of mutually disjoint hexagons of $F_n$ is a sextet pattern if $F_n$ has a perfect matching which alternates on and off each hexagon in $\mathcal H$. The maximum cardinality of sextet patterns of $F_n$ is the Clar number of $F_n$. It was shown that the Clar number is no more than $\lfloor\frac {n-12} 6\rfloor$. Many fullerenes with experimental evidence attain the upper bound, for instance, $\text{C}_{60}$ and $\text{C}_{70}$. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal $\frac{n-12} 6$. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including $\text{C}_{60}$, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.

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