Higher matching complexes of complete graphs and complete bipartite graphs

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