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 ∈V(G),max{dG(u),dG(v)}≥bn/a+b.Moreover,we show that this result is best possible in some sense.