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.
Highlights
It has been known for a long time that the stability of several classes of organic compounds is, to a great degree, determined by the presence and abundance of resonant electronic structures.A natural way to represent such structures is via perfect matchings in the corresponding graphs, and the general rule for polycyclic aromatic hydrocarbons and other conjugated systems is that the stability of a compound increases with the number of perfect matchings in the corresponding graph.The study of matchings in fullerene graphs begun almost as soon as the fullerene structures were first observed
It is well-known that all fullerene graphs have perfect matchings and that all fullerene graphs contain some small connected graphs as nice subgraphs
That answers in affirmative a question posed in a recent paper on nice pairs of odd cycles in fullerene graphs
Summary
It has been known for a long time that the stability of several classes of organic compounds is, to a great degree, determined by the presence and abundance of resonant electronic structures. The study of matchings in fullerene graphs begun almost as soon as the fullerene structures were first observed. It turned out that the number of perfect matchings is not so decisive for the stability of fullerene structures as for the conjugated compounds [1]. It was observed that each fullerene graph contains two disjoint odd cycles whose union is a nice subgraph, and it was investigated whether one or both of those odd cycles could be pentagonal faces. The aim of this contribution is to settle one of the open problems listed at the end of that paper. We list some unsolved issues and indicate a couple of possible topics for future research
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