A result on fractional $k$-deleted graphs

Abstract

Let $k\geq 2$ be an integer, and let $G$ be a graph of order $n$ with $n\geq4k-5$. A graph $G$ is a fractional $k$-deleted graph if there exists a fractional $k$-factor after deleting any edge of $G$. The binding number of $G$ is defined as 26741 {\operatorname {bind}} (G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. 26741 In this paper, it is proved that if ${\operatorname {bind}} (G)>\frac{(2k-1)(n-1)}{k(n-2)}$, then $G$ is a fractional $k$-deleted graph. Furthermore, it is shown that the result in this paper is best possible in some sense.