Number of Matchings of Low Order in (4,6)-Fullerene Graphs

We obtain the formulae for the numbers of 4-matchings and 5-matchings in terms of the number of hexagonal faces in (4, 6)-fullerene graphs by studying structural classification of 6-cycles and some local structural properties, which correct the corresponding wrong results published. Furthermore, we obtain a formula for the number of 6-matchings in tubular (4, 6)-fullerenes in terms of the number of hexagonal faces, and a formula for the number of 6-matchings in the other (4,6)-fullerenes in terms of the numbers of hexagonal faces and dual-squares.Comment: This article was already published in 2017 in MATCH Commun. Math. Comput. Chem. We are uploading it to arXiv for readers' convenienc

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