The Clar covering polynomial of hexagonal systems with an application to chromatic polynomials (Q1366789)

A hexagonal system is a finite connected plane graph with no cut vertices in which every interior face is bounded by a regular hexagon of side length 1. A Clar cover \(C\) of a hexagonal system \(H\) with a perfect matching is a spanning subgraph of \(H\) each component of which is either a hexagon or an edge. The Clar number is the maximum number of hexagons for all such covers, and the Clar covering polynomial \(P(H,w)\) of \(H\) is defined by \(P(H,w)= \sum_r\sigma(H, r)w^r\), where \(\sigma(H, r)\) is the number of Clar covers having precisely \(r\) hexagons. Hexagonal systems are Clar cover equivalent if they have the same polynomial, and a system is Clar cover unique if no other system has the same polynomial. The paper studies the relation between Clar covering polynomials and chromatic polynomials and also establishes the Clar cover equivalence or uniqueness of certain systems.