Abstract Given an arbitrary sequence of non-negative integers λ → = ( λ 1 , … , λ n ) and a graph G with vertex set { v 1 , … , v n } , the bounded degree complex, denoted as BD λ → ( G ) , is a simplicial complex whose faces are the subsets H ⊆ E ( G ) such that for each i ∈ { 1 , … , n } , the degree of vertex v i in the induced subgraph G [ H ] is at most λ i . When λ i = k for all i , the bounded degree complex BD λ → ( G ) is called the k -matching complex, denoted as 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 ≥ 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.

中文翻译：

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森林的有界度复数

摘要 给定一个任意的非负整数序列 λ → = ( λ 1 , … , λ n ) 和一个顶点集 { v 1 , … , vn } 的图 G ，有界度复数，记为 BD λ → ( G ) , 是一个单纯复形，其面是子集 H ⊆ E ( G ) 使得对于每个 i ∈ { 1 , … , n} ，在归纳子图 G [ H ] 中顶点 vi 的度数至多为 λ i 。当对所有 i λ i = k 时，有界度复数 BD λ → ( G ) 称为 k 匹配复数，记为 M k ( G ) 。在本文中，我们确定了森林有界度复合体的同伦类型。特别是，我们证明，对于所有 k ≥ 1，毛毛虫图的 k 匹配复合体要么是可收缩的，要么是等效于球体楔形的同伦，从而证明了 Vega 的猜想（2019，猜想 7.3）。