Bounded degree complexes of forests

Abstract

Given an arbitrary sequence of non-negative integers λ→=(λ1,…,λn) and a graph G with vertex set {v1,…,vn}, the bounded degree complex, denoted as BDλ→(G), is a simplicial complex whose faces are the subsets H⊆E(G) such that for each i∈{1,…,n}, the degree of vertex vi in the induced subgraph G[H] is at most λi. When λi=k for all i, the bounded degree complex BDλ→(G) is called the k-matching complex, denoted as Mk(G).In this article, we determine the homotopy type of bounded degree complexes of forests. In particular, we show that, for all k≥1, the k-matching complexes of caterpillar graphs are either contractible or homotopy equivalent to a wedge of spheres, thereby proving a conjecture of Vega (2019, Conjecture 7.3). 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.