Binding Number and Fractional $k$-Factors of Graphs

Abstract

Let $G$ be a graph with vertex set $V(G)$, and let $k\ge 2$ be an integer. A spanning subgraph $F$ of $G$ is called a fractional $k$-factor if $d_G^h(x)=k$ for all $x\in V(G)$, where $d_G^h(x)=\sum _{e\in E_x}h(e)$ is the fractional degree of $x\in V(F)$ with $E_x=\{e:e=xy,e\in E(G)\}$. The binding number $bind(G)$ is defined as follows:

$$bind(G)=min\{\frac{|N_G(X)|}{|X|}:\varnothing \ne X\subseteq V(G),N_G(G)\ne V(G)\}.$$

In this paper, a binding number condition for a graph to have fractional $k$-factors is given.