Binding numbers and f-factors of graphs

AbstractLet 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 Σ(f(x); x ∈ V(G)) ≡ 0 (mod 2). 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)2a, then G has an f-factor; (ii) If the minimum degree of G is greater than (bn − 2)(a + b), and n ≥(a + b)2a, then G has an f-factor