Components of domino tilings under flips in quadriculated cylinder and torus

TL;DR

This paper investigates the structure of domino tilings on quadriculated cylinders and tori, revealing connectivity properties of flip graphs and analyzing forcing numbers, with implications for understanding tiling configurations on these surfaces.

Contribution

It demonstrates that the flip graph for domino tilings on certain cylinders is connected, while on tori it splits into two components, and characterizes the forcing numbers of all tilings.

Findings

Flip graph of 2m x (2n+1) quadriculated cylinder is connected.

Flip graph of 2m x (2n+1) quadriculated torus has two isomorphic components.

Forcing numbers form an integer interval with maximum (n+1)m.

Abstract

In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the set of all tilings of $R$ such that two tilings are adjacent if we change one to another by a flip (a $90^{\circ}$ rotation of a pair of side-by-side dominoes). It is well-known that $\mathcal{T}(R)$ is connected when $R$ is simply connected. By using graph theoretical approach, we show that the flip graph of $2m\times(2n+1)$ quadriculated cylinder is still connected, but the flip graph of $2m\times(2n+1)$ quadriculated torus is disconnected and consists of exactly two isomorphic components. For a tiling $t$, we associate an integer $f(t)$, forcing number, as the minimum number of dominoes in $t$ that is contained in no other tilings. As an application, we obtain that the forcing numbers of all tilings in $2m\times (2n+1)$ quadriculated cylinder and torus form respectively an integer interval whose maximum value is $(n+1)m$.