Matching complexes of trees and applications of the matching tree algorithm

TL;DR

This paper investigates the topology of matching complexes for specific tree families, providing explicit formulas and bounds, and introduces the matching tree algorithm to analyze connectivity in honeycomb graphs.

Contribution

It offers explicit formulas for caterpillar graphs, connectivity bounds for perfect binary trees, and applies the matching tree algorithm to honeycomb graphs.

Findings

Matching complexes of caterpillar graphs have explicit sphere counts.

Connectivity bounds are established for perfect binary trees.

The matching tree algorithm helps analyze honeycomb graph connectivity.

Abstract

A matching complex of a simple graph $G$ is a simplicial complex with faces given by the matchings of $G$. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa showed that matching complexes of forests are contractible or homotopy equivalent to a wedge of spheres. We study two specific families of trees. For caterpillar graphs, we give explicit formulas for the number of spheres in each dimension and for perfect binary trees we find a strict connectivity bound. We also use a tool from discrete Morse theory called the \textit{Matching Tree Algorithm} to study the connectivity of honeycomb graphs, partially answering a question raised by Jonsson.