Binding Number and Fractional \(k\)-Factors of Graphs

Sizhong Zhou1, Zurun Xu2
1School of Mathematics and Physics Jiangsu University of Science and Technology Mengxi Road 2, Zhenjiang, Jiangsu 212003, P. R. China
2 School of Science, China University of Mining and Technology Xuzhou, Jiangsu 221008, P. R. China

Abstract

Let \(G\) be a graph with vertex set \(V(G)\), and let \(k \geq 2\) 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:

\[bind(G) = \min\left\{\frac{|N_G(X)|}{|X|} :\varnothing \neq X \subseteq V(G), N_G(G) \neq V(G)\right\}.\]

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