Sufficient Conditions for a Graph to Have All [ a , b ]-Factors and ( a , b )-Parity Factors

Let G be a graph with vertex set V and let b>a be two positive integers. We say that G has all [ a , b ]-factors if G has an h -factor for every h: V→ℕ such that a ≤ h(v) ≤ b for every v∈ V and ∑ _v∈ Vh(v)≡ 0 2 . A spanning subgraph F of G is called an ( a , b )-parity factor, if d_F (v) ≡ a ≡ b (mod 2) and a ≤ d_F (v) ≤ b for all v ∈ V . In this paper, we have developed sufficient conditions for the existence of all [ a , b ]-factors and ( a , b )-parity factors of G in terms of the independence number and connectivity of G . This work extended an earlier result of Nishimura (J Graph Theory 13: 63–69, 1989). Furthermore, we show that these results are best possible in some cases.