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).