  For $r\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is
a simplicial complex whose faces are the subsets $H \subseteq E(G)$ 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 $K_{n,n}$
is homotopy equivalent to a sphere of dimension $(n-1)^2-1$.
