A Degree Condition for Graphs Having All (a, b)-parity Factors

Let a and b be positive integers such that a ≤ b and a ≡ b (mod 2). We say that G has all (a, b)-parity factors if G has an h-factor for every function h: V(G) → {a, a + 2, ⋯, b − 2, b} with b∣V(G)∣ even and h(v) ≡ b (mod 2) for all v ∈ V(G). In this paper, we prove that every graph G with n ≥ 2(b + 1)(a + b) vertices has all (a, b)-parity factors if δ(G) ≥ (b² − b)/a, and for any two nonadjacent vertices \(u,\,v\, \in \,V\,(G),\,\max \{{d_G}(u),\,{d_G}(v)\} \, \ge {{bn} \over {a + b}}\) $$u,v\in V(G),\max \{d_G(u),d_G(v)\}\ge \frac{bn}{a+b}$$ . Moreover, we show that this result is best possible in some sense.