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 $F$.Comment: 11 pages, 9 figure

Fullerenes with the maximum Clar number

The Clar number of a fullerene is the maximum number of independent resonant hexagons in the fullerene. It is known that the Clar number of a fullerene with n vertices is bounded above by [n/6]-2. We find that there are no fullerenes whose order n is congruent to 2 modulo 6 attaining this bound. In other words, the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is bounded above by [n/6]-3. Moreover, we show that two experimentally produced fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a graph-theoretical characterization for fullerenes, whose order n is congruent to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3 (respectively, [n/6]-2)

Fullerene graphs have exponentially many perfect matchings

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.Comment: 7 pages, 3 figure

2-Resonant fullerenes

A fullerene graph $F$ is a planar cubic graph with exactly 12 pentagonal faces and other hexagonal faces. A set $\mathcal{H}$ of disjoint hexagons of $F$ is called a resonant pattern (or sextet pattern) if $F$ has a perfect matching $M$ such that every hexagon in $\mathcal{H}$ is $M$-alternating. $F$ is said to be $k$-resonant if any $i$ ($0\leq i\leq k$) disjoint hexagons of $F$ form a resonant pattern. It was known that each fullerene graph is 1-resonant and all 3-resonant fullerenes are only the nine graphs. In this paper, we show that the fullerene graphs which do not contain the subgraph $L$ or $R$ as illustrated in Fig. 1 are 2-resonant except for the specific eleven graphs. This result implies that each IPR fullerene is 2-resonant.Comment: 34 pages, 25 figure