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Let  G  be a graph with vertex set  V  and let    $$b&gt;a$$  b &gt; a be two positive integers. We say that  G  has all [ a ,  b ]-factors if  G  has an  h -factor for every    $$h: V\rightarrow {\mathbb {N}}$$  h : V → N such that    $$a \le h(v) \le b$$  a ≤ h ( v ) ≤ b for every    $$v\in V$$  v ∈ V and    $$\sum _{v\in V}h(v)\equiv 0\pmod 2$$  ∑ v ∈ V h ( v ) ≡ 0 ( mod 2 ) . A spanning subgraph  F  of  G  is called an ( a ,  b )-parity factor, if    $$d_F (v) \equiv a \equiv b $$  d F ( v ) ≡ a ≡ b (mod 2) and    $$a \le d_F (v) \le b$$  a ≤ d F ( v ) ≤ b for all    $$v \in V$$  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.

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