# The cubical matching complex revisited

## TL;DR

This paper revisits the cubical matching complex, clarifies its topological properties, extends its definition to broader classes of graphs, and establishes a unique linear relation among tiling counts for domino and square tilings.

## Contribution

It corrects and clarifies previous claims about the collapsibility of cubical matching complexes, extends their definition to non-bipartite graphs, and proves a unique linear relation among tiling counts.

## Key findings

- Links in the complexes are suspensions up to homotopy.
- Cubical matching complexes for non-bipartite graphs are contractible or unions of contractible complexes.
- The relation $f_0 - f_1 + f_2 - ... = 1$ is the only linear relation among tiling counts.

## Abstract

Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed that this complex is collapsible. We point out to an oversight in his proof and explain why these complexes can be the union of collapsible complexes. Also, we prove that all links in these complexes are suspensions up to homotopy. Furthermore, we extend the definition of a cubical matching complex to planar graphs that are not necessarily bipartite, and show that these complexes are either contractible or a disjoint union of contractible complexes. For a simple connected region that can be tiled with dominoes ($2\times 1$ and $1\times 2$) and $2\times 2$ squares, let $f_i$ denote the number of tilings with exactly $i$ squares. We prove that $f_0-f_1+f_2-f_3+\cdots=1$ (established by Ehrenborg) is the only linear relation for the numbers $f_i$.