All Pairs of Pentagons in Leapfrog Fullerenes Are Nice

Abstract

A subgraph H of a graph G with perfect matching is nice if G−V(H) has perfect matching. It is well-known that all fullerene graphs have perfect matchings and that all fullerene graphs contain some small connected graphs as nice subgraphs. In this contribution, we consider fullerene graphs arising from smaller fullerenes via the leapfrog transformation, and show that in such graphs, each pair of (necessarily disjoint) pentagons is nice. That answers in affirmative a question posed in a recent paper on nice pairs of odd cycles in fullerene graphs.

Figures of Graph Partitioning by Counting, Sequence and Layer Matrices

“…Fullerenes are structures [84] with a high symmetry, stabilized by resonance, in which the difference between different positions (atoms) are very small and are of interest for their reactivity and functionalization [85]. Following this idea, of interest is identifying, visually if possible, different equivalent positions in the structures.…”

Nice pairs of pentagons in chamfered fullerenes

“…In [1] it is proven that all pairs of pentagons in Leapfrog Fullerenes are nice. We extened this result for chamfered fullerenes (Fullerenes obtained the chamfering transformation from smaller fullerenes) after the fashion of [1].…”