Odd cycle transversals and independent sets in fullerene graphs

TL;DR

This paper proves bounds on how to make fullerene graphs bipartite by edge deletion and on their independence number, confirming conjectures and characterizing extremal cases, with implications for eigenvalues.

Contribution

It establishes sharp bounds on edge deletions for bipartiteness and on independence numbers in fullerene graphs, confirming two conjectures and characterizing extremal graphs.

Findings

Every fullerene graph can be made bipartite by deleting at most sqrt{12n/5} edges.

Fullerene graphs have an independent set of size at least n/2 - sqrt{3n/5}.

The bounds are sharp and extremal graphs are characterized.

Abstract

A fullerene graph is a cubic bridgeless plane graph with all faces of size 5 and 6. We show that that every fullerene graph on n vertices can be made bipartite by deleting at most sqrt{12n/5} edges, and has an independent set with at least n/2-sqrt{3n/5} vertices. Both bounds are sharp, and we characterise the extremal graphs. This proves conjectures of Doslic and Vukicevic, and of Daugherty. We deduce two further conjectures on the independence number of fullerene graphs, as well as a new upper bound on the smallest eigenvalue of a fullerene graph.