Higher matching complexes of complete graphs and complete bipartite graphs

Abstract

For r≥1, the r-matching complex of a graph G, denoted Mr(G), is a simplicial complex whose faces are the subsets H⊆E(G) of the edge set of G such that the degree of any vertex in the induced subgraph G[H] is at most r. In this article, we give a closed form formula for the homotopy type of the (n−2)-matching complex of complete graph on n vertices. We also prove that the (n−1)-matching complex of complete bipartite graph Kn,n is homotopy equivalent to a sphere of dimension (n−1)2−1.