Nice pairs of odd cycles in fullerene graphs

Abstract

Let G be a graph with a perfect matching. A subgraph H of G is nice if $$G - V(H)$$ still has a perfect matching. In a chemical context, nice subgraphs of molecular graphs serve as mathematical models of addition patterns in the corresponding molecules such that the rest of the molecule still has a resonant structure. In this contribution we start from the fact that each fullerene graph has a nice pair of disjoint odd cycles and investigate when one or both cycles in such pairs can be chosen to be pentagons. Along the way we completely settle the analogous problem for closely related generalized fullerenes with only triangular and hexagonal faces. We also report some computational results for small fullerenes and list some open problems.