  In data transmission networks, the availability of data transmission is
equivalent to the existence of the fractional factor of the corresponding graph
which is generated by the network. Research on the existence of fractional
factors under specific network structures can help scientists design and
construct networks with high data transmission rates. 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 after removing any $n$ vertices of $G$, the
resulting graph of $G$ is a fractional $(g,f)$-covered graph. In this paper, we
verify that if a graph $G$ of order $p$ satisfies
$p\geq\frac{(a+b-1)(a+b-2)+(a+d)n+1}{a+d}$,
$\delta(G)\geq\frac{(b-d-1)p+(a+d)n+a+b+1}{a+b-1}$ and
$\delta(G)>\frac{(b-d-2)p+2\alpha(G)+(a+d)n+1}{a+b-2}$, then $G$ is a
fractional $(g,f,n)$-critical covered graph, where $g,f:V(G)\rightarrow Z^{+}$
be two functions such that $a\leq g(x)\leq f(x)-d\leq b-d$ for all $x\in V(G)$,
which is a generalization of Zhou's previous result [S. Zhou, Some new
sufficient conditions for graphs to have fractional $k$-factors, International
Journal of Computer Mathematics 88(3)(2011)484--490].

Combinatorics