2-Resonant fullerenes

TL;DR

This paper investigates the resonant properties of fullerene graphs, establishing conditions under which they are 2-resonant, and identifying exceptions, thereby advancing understanding of their perfect matching structures.

Contribution

It proves that fullerene graphs without certain subgraphs are 2-resonant, except for eleven specific graphs, extending known results on fullerene resonance.

Findings

Most fullerene graphs without subgraphs L or R are 2-resonant.

All IPR fullerenes are 2-resonant.

Identifies eleven exceptions to 2-resonance in certain fullerene graphs.

Abstract

A fullerene graph $F$ is a planar cubic graph with exactly 12 pentagonal faces and other hexagonal faces. A set $\mathcal{H}$ of disjoint hexagons of $F$ is called a resonant pattern (or sextet pattern) if $F$ has a perfect matching $M$ such that every hexagon in $\mathcal{H}$ is $M$-alternating. $F$ is said to be $k$-resonant if any $i$ ($0\leq i\leq k$) disjoint hexagons of $F$ form a resonant pattern. It was known that each fullerene graph is 1-resonant and all 3-resonant fullerenes are only the nine graphs. In this paper, we show that the fullerene graphs which do not contain the subgraph $L$ or $R$ as illustrated in Fig. 1 are 2-resonant except for the specific eleven graphs. This result implies that each IPR fullerene is 2-resonant.