The Clar covering polynomial of hexagonal systems. III
[For Part I see Discrete Appl. Math. 69, No. 1-2, 147-167 (1996; Zbl 0859.05070).] Let \(H\) be a hexagonal system with a perfect matching. A Clar cover of \(H\) is a spanning subgraph each component of which is either a hexagon or an edge. The set of hexagons in a Clar cover is a resonant pattern of \(H\) and a resonant pattern with the maximum number \(C(H)\) of hexagons is a Clar formula; \(C(H)\) is the Clar number. The sextet polynomial \(B(H,x)= \sum_n s(H, n)x^n\) where \(s(H,n)\) is the number of resonant patterns with \(n\) hexagons. The Clar covering polynomial \(P(H, w)= \sum_n\sigma(H, n)w^n\) where \(\sigma(H, n)\) is the number of Clar covers with \(n\) hexagons. In this paper a relation is etablished between these two polynomials, a lower bound for the Clar number is obtained and some properties of the Clar covering polynomial are deduced.
scientific article; zbMATH DE number 1421662