Perfect matchings of (4,6)-fullerenes with largest forcing number

Clar number (or resonant number) is a thoroughly investigated parameter of plane graphs emerging from mathematical chemistry to measure stability of some organic molecules. It was shown that the Clar number of a $(4,6)$-fullerene graph $G$, a plane cubic graph with only hexagonal and quadrilateral faces, is equal to its maximum forcing number $F\left(G\right)$. Let $M$ be any perfect matching of $G$ attaining the maximum forcing number. We use $C(G,M)$ and $C(G,M)$ to denote two largest sets of disjoint $M$-alternating cycles and $M$-alternating facial cycles respectively. Then $F\left(G\right)=\left|C(G,M)\right|\ge \left|C(G,M)\right|$. In this paper, we consider when $\left|C(G,M)\right|=\left|C(G,M)\right|$ holds. First we show that every cycle in $C(G,M)$ has length 4, 6, 8 or 12. Then we construct two types of tubular $(4,6)$-fullerene graphs $G_1$ and $G_2$ so that there is a $C(G,M)$ containing a 12-cycle if and only if $G\in G_1$, and there is a $C(G,M)$ containing a 8-cycle if and only if $G\in G_2$. As a consequence, we obtain the following three equivalent statements: (i) Each $C(G,M)$ consists of 6-cycles and 4-cycles; (ii) $\left|C(G,M)\right|=\left|C(G,M)\right|$; (iii) $G\notin G_1\cup G_2$.