Nice pairs of disjoint pentagons in fullerene graphs

Abstract

A subgraph H of a graph G with a perfect matching is called nice, if $$G-V(H)$$ has a perfect matching. Very recently, Doslic (JMC 58:2204-2222, 2020) pointed out that every fullerene graph contains a nice pair of disjoint odd cycles and proposed a series of problems on whether such a pair of odd cycles can be chosen as pentagons. In this paper, we show that any fullerene graph always contains a nice pair of disjoint pentagons, which solves completely Problems 1 and 2. Further, we prove that if a fullerene graph satisfies that each pentagon is adjacent to at most two other pentagons, then any pair of disjoint pentagons is nice except for only one fullerene graph on 36 vertices. As a consequence, for the IPR fullerenes (any two pentagons are disjoint), the result also holds, which thus answers positively Problems 4 and 6.