Equivalence between Clar covering polynomials of single zigzag chains and tiling polynomials of 2 x n rectangles

Abstract

This paper offers a formal explanation of a rather puzzling and surprising equivalence between the Clar covering polynomials of single zigzag chains and the tiling polynomials of 2 x n rectangles for tilings using 1 x 2, 2 x 1 and 2 x 2 tiles. It is demonstrated that the set of Clar covers of single zigzag chains N(n - 1) is isomorphic to the set of filings of a 2 x n rectangle. In particular, this isomorphism maps Clar covers of N(n - 1) with k aromatic sextets to tilings of a 2 x n rectangle using k square 2 x 2 tiles. The proof of this fact is an application of the recently introduced interface theory of Clar covers. The existence of a similar relationship between the Clar covers of more general benzenoid structures and more general tilings of rectangles remains an interesting open problem in chemical graph theory. (C) 2018 Elsevier B.V. All rights reserved.