Bounded degree complexes of forests

TL;DR

This paper characterizes the homotopy types of bounded degree complexes, especially k-matching complexes of forests and caterpillar graphs, proving they are either contractible or wedge of spheres, confirming a conjecture.

Contribution

It determines the homotopy types of bounded degree complexes of forests, especially proving a conjecture for k-matching complexes of caterpillar graphs.

Findings

k-matching complexes of caterpillar graphs are contractible or wedge of spheres

Homotopy types of bounded degree complexes of forests are characterized

Closed form formulas for certain caterpillar graphs' complexes

Abstract

Given an arbitrary sequence of non-negative integers $\vec{\lambda}=(\lambda_1,\dots,\lambda_n)$ and a graph $G$ with vertex set $\{v_1,\dots,v_n\}$, the bounded degree complex, denoted $\text{BD}^{\vec{\lambda}}(G)$, is a simplicial complex whose faces are the subsets $H\subseteq E(G)$ such that for each $i \in \{1,\dots,n\}$, the degree of vertex $v_i$ in the induced subgraph $G[H]$ is at most $\lambda_i$. When $\lambda_i=k$ for all $i$, the bounded degree complex $\text{BD}^{\vec{\lambda}}(G)$ is called the $k$-matching complex, denoted $M_k(G)$. In this article, we determine the homotopy type of bounded degree complexes of forests. In particular, we show that, for all $k\geq 1$, the $k$-matching complexes of caterpillar graphs are either contractible or homotopy equivalent to a wedge of spheres, thereby proving a conjecture of Julianne Vega \cite[Conjecture 7.3]{Vega19}. We also give a closed form formula for the homotopy type of the bounded degree complexes of those caterpillar graphs in which every non-leaf vertex is adjacent to at least one leaf vertex.