Binding numbers and f-factors of graphs

Let G be a connected graph of order n, a and b be integers such that 1 ≤ a ≤ b and 2 ≤ b, and f : V (G) → {a, a + 1, . . . , b} be a function such that ∑ (f(x);x ∈ V (G)) ≡ 0(mod2). We prove the following two results: (i) If the binding number of G is greater than (a+b−1)(n−1)/(an−(a+b)+3) and n ≥ (a+b)2/a, then G has an f factor; (ii) if the minimum degree of G is greater than (bn−2)/(a+b), and n ≥ (a+ b)2/a, then G has an f -factor.