Independence number,connectivity, and r-factors

We show that if r ? 1 is an odd integer and G is a graph with |V(G)| even such that k(G) ? (r + 1)²/2 and (r + 1)²α(G) ? 4rk(G), then G has an r-factor; if r ? 2 is even and G is a graph with k(G) ? r(r + 2)/2 and (r + 2)α(G) ? 4k(G), then G has an r-factor (where k(G) and α(G) denote the connectivity and the independence number of G, respectively).