Equivalence between Clar covering polynomials of single zigzag chains and tiling polynomials of 2×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 × n rectangles for tilings using 1  ×  2, 2  ×  1 and 2  ×  2 tiles. It is demonstrated that the set of Clar covers of single zigzag chains N ( n − 1 ) is isomorphic to the set of tilings of a 2 × n rectangle. In particular, this isomorphism maps Clar covers of N ( n − 1 ) with k aromatic sextets to tilings of a 2 × n rectangle using k square 2  ×  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.

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