Extremal fullerene graphs with the maximum Clar number

Abstract

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.

35 pages, 43 figures

Cited by in corpus (1)

• On k-resonant fullerene graphs