Abstract
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⋅
2
⌊
n
20
⌋
perfect matchings, where
n
is the order of
F
.