A degree condition for fractional ( g, f, n )-critical covered graphs

A graph $G$ is called a fractional $(g,f)$-covered graph if for any $e\in E(G)$, $G$ admits a fractional $(g,f)$-factor covering $e$. A graph $G$ is called a fractional $(g,f,n)$-critical covered graph if for any $W\subseteq V(G)$ with $|W|=n$, $G-W$ is a fractional $(g,f)$-covered graph. In this paper, we demonstrate that a graph $G$ of order $p$ is a fractional $(g,f,n)$-critical covered graph if $p\geq\frac{(a+b)(a+b+n+1)-(b-m)n+2}{a+m}$, $\delta(G)\geq\frac{(b-m)(b+1)+2}{a+m}+n$ and for every pair of nonadjacent vertices $u$ and $v$ of $G$, $\max\{d_G(u),d_G(v)\}\geq\frac{(b-m)p+(a+m)n+2}{a+b}$, where $g$ and $f$ are integer-valued functions defined on $V(G)$ satisfying $a\leq g(x)\leq f(x)-m\leq b-m$ for every $x\in V(G)$.