$(1,f)$-Factors of Graphs with Odd Property

Abstract

Let $G$ be a graph and $f:V(G)\rightarrow \{1,2,3,4,\ldots \}$ be a function. We denote by $odd(G)$ the number of odd components of $G$. We prove that if $odd(G-X)\le \sum _{x\in X}f(x)$ for all $X\subset V(G)$, then $G$ has a $(1,f)$-factor $F$ such that, for every vertex $v$ of $G$, if $f(v)$ is even, then $\deg _F(v)\in \{1,3,\ldots ,f(v)-1,f(v)\}$, and otherwise $\deg _F(v)\in \{1,3, \ldots , f(v)\}$. This theorem is a generalization of both the $(1,f)$-odd factor theorem and a recent result on $\{1,3, \ldots , 2n-1,2n\}$-factors by Lu and Wang. We actually prove a result stronger than the above theorem.