Extremal fullerene graphs with the maximum Clar number

TL;DR

This paper characterizes extremal fullerene graphs with maximum Clar number, identifies all such graphs with 60 vertices including C60, and constructs all these extremal structures.

Contribution

It provides a complete characterization of extremal fullerene graphs with maximum Clar number and explicitly constructs all such graphs with 60 vertices.

Findings

Exactly 18 fullerene graphs with 60 vertices attain the maximum Clar number.

C60 is among the extremal fullerene graphs with maximum Clar number.

The paper constructs all extremal fullerene graphs with 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.