Perfect Matching Complexes of Honeycomb Graphs

TL;DR

This paper investigates the topological structure of perfect matching complexes in honeycomb graphs, revealing their homotopy types and contractibility properties using discrete Morse theory.

Contribution

It characterizes the homotopy types of perfect matching complexes of honeycomb graphs for specific parameters, providing new insights into their topological structure.

Findings

For $k=1$, the complex is contractible unless $n extgreater{} m=2$, then it is homotopy equivalent to a sphere.

The complex $ ext{M}_p(H_{2 imes 2 imes 2})$ is homotopy equivalent to a wedge of two 3-spheres.

Proofs utilize discrete Morse theory to analyze the complexes.

Abstract

The {\em perfect matching complex} of a graph is the simplicial complex on the edge set of the graph with facets corresponding to perfect matchings of the graph. This paper studies the perfect matching complexes, $\mathcal{M}_p(H_{k \times m\times n})$, of honeycomb graphs. For $k = 1$, $\mathcal{M}_p(H_{1\times m\times n})$ is contractible unless $n\ge m=2$, in which case it is homotopy equivalent to the $(n-1)$-sphere. Also, $\mathcal{M}_p(H_{2\times 2\times 2})$ is homotopy equivalent to the wedge of two 3-spheres. The proofs use discrete Morse theory.