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arXiv · 2307.08332

Components of domino tilings under flips in quadriculated tori

Qianqian Liu·Yaxian Zhang·Heping Zhang
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Abstract

In a region R consisting of unit squares, a (domino) tiling is a collection of dominoes (the union of two adjacent squares) which pave fully the region. The flip graph of R is defined on the set of all tilings of R where two tilings are adjacent if we change one from the other by a flip (a 90-degree rotation of a pair of side-by-side dominoes). If R is simply-connected, then its flip graph is connected. By using homology and cohomology, Saldanha, Tomei, Casarin and Romualdo obtained a criterion to decide if two tilings are in the same component of flip graph of quadriculated surface. By a graph-theoretic method, we obtain that the flip graph of a non-bipartite quadriculated torus consists of two isomorphic components. As an application, we obtain that the forcing numbers of all perfect matchings of each non-bipartite quadriculated torus form an integer-interval. For a bipartite quadriculated torus, the components of the flip graph is more complicated, and we use homology to obtain a general lower bound for the number of components of its flip graph.

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Qianqian Liu, Yaxian Zhang, Heping Zhang. 2023-07-17. Components of domino tilings under flips in quadriculated tori. https://arxiv.org/abs/2307.08332

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