### 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 language | English |
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Pages (from-to) | 297-303 |

Number of pages | 7 |

Journal | Discrete Applied Mathematics |

Volume | 243 |

DOIs | |

State | Published - 10 Jul 2018 |

### Keywords

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