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(mod2). A spanning subgraph F of G is called an (a, b)-parity factor, if dF(v) ≡ a≡ b (mod 2) and a≤ dF(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.