Components of domino tilings under flips in quadriculated tori

TL;DR

This paper investigates the structure of flip graphs of domino tilings on quadriculated tori, revealing their component structure and properties using homology, and establishing bounds on the number of components and forcing numbers.

Contribution

It proves that non-bipartite quadriculated tori have flip graphs with two isomorphic components and characterizes the forcing numbers of their perfect matchings.

Findings

Non-bipartite quadriculated tori have flip graphs with two isomorphic components.

Forcing numbers of perfect matchings in non-bipartite tori form an integer interval.

Homology provides bounds on the number of components in bipartite tori flip graphs.

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.