Odd Cycle Transversals and Independent Sets in Fullerene Graphs

A fullerene graph is a cubic bridgeless plane graph with all faces of size 5 and 6. We show that every fullerene graph on n vertices can be made bipartite by deleting at most √ 12n/5 edges, and has an independent set with at least n/2 − √ 3n/5 vertices. Both bounds are sharp, and we characterise the extremal graphs. This proves conjectures of Došlić and Vukičević, 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.