Components of domino tilings under flips in quadriculated tori

TL;DR

This work clarifies how domino tiling flip graphs on quadriculated tori decompose by bipartiteness: non-bipartite tori yield exactly two isomorphic components, while bipartite tori admit a richer, homology-informed component structure that provides lower bounds on component counts. The authors leverage resonance graphs and homology, introducing $I$-cycles and flux concepts, to prove the two-component phenomenon for all non-bipartite tori and to derive lower bounds for bipartite cases. A key application is that forcing spectra are continuous (integer-interval) for non-bipartite tori, contrasted with possible gaps in bipartite cases, demonstrated computationally. Overall, the paper connects combinatorial tiling moves, topological invariants, and dimer theory to yield precise descriptions of flip-graph components and forcing spectra on toroidal grids.

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.

Figures (13)

• Figure 1: Quadriculated torus $T(4,8,2)$ (left), and labels of its vertices (right).
• Figure 2: New labels of $T(3,12,4)$ according to $I$-cycles
• Figure 3: Quadriculated torus $T(4,9,6)$, a dual representation of $T(3,12,4)$.
• Figure 4: Some squares of $T(5,8,4)$.
• Figure 5: Description of the proof of Lemma \ref{['lem61']}.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• proof
• Lemma 2.5
• proof
• Corollary 2.6