A degree condition for graphs being fractional (a,b,k)-critical covered

Abstract

A graph $G$ is fractional $[a,b]$-covered if for any $e\in E(G)$, $G$ possesses a fractional $[a,b]$-factor including $e$. A graph
$G$ is fractional $(a,b,k)$-critical covered if $G-Q$ is fractional $[a,b]$-covered for any $Q\subseteq V(G)$ with $|Q|=k$. In this paper, we verify that a graph $G$ of order $n$ is fractional $(a,b,k)$-critical covered if $n\geq\frac{(a+b)((2r-3)a+b+r-2)+bk+2}{b}$, $\delta(G)\geq(r-1)(a+1)+k$ and
$$
\max\{d_G(w_1),d_G(w_2),\cdots,d_G(w_r)\}\geq\frac{an+bk+2}{a+b}
$$
for every independent vertex subset $\{w_1,w_2,\cdots,w_r\}$ of $G$. Our main result is an improvement of the previous result
[S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional $(a,b,k)$-critical covered graphs, Information Processing Letters 152(2019)105838].