On the anti-forcing number of fullerene graphs

TL;DR

This paper investigates the anti-forcing number in fullerene graphs, establishing a lower bound of four, providing a construction method for such fullerenes, and demonstrating the existence of fullerenes with specific anti-forcing numbers for various sizes.

Contribution

It proves that all fullerenes have an anti-forcing number of at least four and offers a construction method for those achieving this bound, also identifying exceptions.

Findings

All fullerenes have an anti-forcing number ≥ 4.

Constructive procedure for fullerenes with anti-forcing number 4.

Existence of fullerenes with specific anti-forcing numbers for given sizes.

Abstract

The anti-forcing number of a connected graph $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching. In this paper, we show that the anti-forcing number of every fullerene has at least four. We give a procedure to construct all fullerenes whose anti-forcing numbers achieve the lower bound four. Furthermore, we show that, for every even $n\geq20$ ($n\neq22,26$), there exists a fullerene with $n$ vertices that has the anti-forcing number four, and the fullerene with 26 vertices has the anti-forcing number five.