Perfect Matching Complexes of Polygonal Line Tilings

TL;DR

This work investigates the topology of the perfect matching complex $\mathcal{M}_p(G)$ for polygonal line tilings and the $(2\\times n)$-grid. It uses discrete Morse theory to construct acyclic matchings and shows that $\mathcal{M}_p(G)$ is either contractible or a wedge of spheres, introducing the notion of bad matchings to analyze extendability to perfect matchings. The authors determine the homotopy type for the $2\\times n$ grid (contractible for odd $n$, and $\\mathbb{S}^k$ when $n=2k+2$ is even), the line tiling of even-sided polygons (contractible), and the odd-sided case (contractible under simple/alternate arrangements; triangles yield parity-dependent wedges of spheres). They further conjecture a general wedge-of-spheres structure for $\\mathcal{G}_{m\\times n}$ and highlight potential links between matching and perfect matching complexes, offering a framework for extending these results to broader grid families.

Abstract

The perfect matching complex of a simple graph $G$ is a simplicial complex having facets (maximal faces) as the perfect matchings of $G$. This article discusses the perfect matching complex of polygonal line tilings and the $\left(2 \times n\right)$-grid graph in particular. We use tools from discrete Morse theory to show that the perfect matching complex of any polygonal line tiling is either contractible or homotopy equivalent to a wedge of spheres. While proving our results, we also characterize all the matchings of $\left(2 \times n\right)$-grid graph that cannot be extended to form a perfect matching.

Figures (20)

• Figure 1: The bad matching $\left\{a,d\right\}$ of $C_6$.
• Figure 2: Extending the matching $\left\{a,c\right\}$ to the perfect matching $\left\{a,c,e\right\}$ of $C_6$.
• Figure 3: Labelling in $2\times n$ Grid Graph
• Figure 4: The set given in X1.
• Figure 5: The set given in X2.

Theorems & Definitions (32)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.1
• Corollary 2.2
• Definition 2.5: deshpande2020higher, jonsson2008simplicial
• Theorem 2.3
• Lemma 3.1