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) 2 α( G ) ⩽ 4 rk ( 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 ) ⩽ 4 k ( G ), then G has an r ‐factor (where k ( G ) and α( G ) denote the connectivity and the independence number of G , respectively).