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

Johanna Langner, Henryk A. Witek*

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Scopus citations

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.

Original languageEnglish
Pages (from-to)297-303
Number of pages7
JournalDiscrete Applied Mathematics
Volume243
DOIs
StatePublished - 10 Jul 2018

Keywords

  • Clar structure
  • Interface
  • Kekulé structure
  • Perfect matching
  • Tiling
  • Zhang–Zhang polynomial

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