A Note on Fractional $(g,f,m)$-Deleted Graphs

Abstract

A graph $G$ is called a fractional $(g,f,m)$-deleted graph if after deleting any $m$ edges, then the resulting graph admits a fractional $(g,f)$-factor. In this paper, we prove that if $G$ is a graph of order $n$, and if $1\le g(x)\le f(x)\le 6$ for any $x\in V(G)$, $\delta (G)\ge \frac{b^2(i-1)}{a}++2m$, $n>\frac{(a+b)(i(a+b)+2m-2)}{a}$ and $|N_G(x_1)\cup N_G(x_2)\cup \cdots \cup N_G(x_i)|\ge \frac{bn}{a+b}$, for any independent set $\{x_1,x_2,\dots ,x_i\}$ of $V(G)$, where $i\ge 2$, then $G$ is a fractional $(g,f,m)$-deleted graph. The result is tight on the neighborhood union condition.