On Cyclic Edge-Connectivity of Fullerenes

A graph is said to be cyclic $k$-edge-connected, if at least $k$ edges must be removed to disconnect it into two components, each containing a cycle. Such a set of $k$ edges is called a cyclic-$k$-edge cutset and it is called a trivial cyclic-$k$-edge cutset if at least one of the resulting two components induces a single $k$-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this article it is shown that a fullerene $F$ containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that $F$ has a Hamilton cycle, and as a consequence at least $15\cdot 2^{\lfloor \frac{n}{20}\rfloor}$ perfect matchings, where $n$ is the order of $F$.