A neighborhood condition for graphs to have [a, b]-factors II

Let a, b, m, and t be integers such that 1 ≤ a < b and 1 ≤ t ≤ [(b - m + 1)/a]. Suppose that G is a graph of order |G| and H is any subgraph of G with the size |E(H)| = m. Then we prove that G has an [a, b]-factor containing all the edges of H if the minimum degree is at least a, |G| > ((a + b)(t(a + b - 1) - 1) + 2m)/b, and |N_G(x₁) ∪... ∪N_G(x_t)| ≥ (a|G| + 2m)/(a + b) for every independent set {x₁,..., x_t} ⊆ V (G). This result is best possible in some sense and it is an extension of the result of H. Matsuda (A neighborhood condition for graphs to have [a, b]-factors, Discrete Mathematics 224 (2000) 289-292).