For integers a and b such that 0 ≤ a ≤ b, a graph G is called an [a, b]−graph if a ≤ dG(x) ≤ b for every vertex x of G and a factor F of a graph is called an [a, b]−factor if a ≤ dF (x) ≤ b for every vertex x of F . We prove the following theorems. Let 0 ≤ l ≤ k ≤ r, 0 ≤ s, 0 ≤ u and 1 ≤ t. Then an [r, r+s]−graph has a [k, k+ t]-factor if ks ≤ rt. Moreover, if (k− l)s+ (r−k)u ≤ (r−1)t, then an ...