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

Abstract

Let $a$ and $b$ be positive integers such that $a\le b$ and $a\equiv b(\mathrm{mod}2)$. We say that $G$ has all $(a,b)$-parity factors if $G$ has an $h$-factor for every function $h:V(G)\to \{a,a+2,\dots ,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b(\mathrm{mod}2)$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\ge 3(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $\delta (G)\ge (b^2-b)/a$, and for any two nonadjacent vertices $u,v\in V(G)$, $\max \{d_G(u),d_G(v)\}\ge \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.