Sufficient conditions for fractional [a, b]-deleted graphs

Abstract

Let a and b be two positive integers with $a\le b$, and let G be a graph with vertex set V(G) and edge set E(G). Let $h:E(G)\rightarrow [0,1]$ be a function. If $a\le \sum \limits _{e\in E_G(v)}{h(e)}\le b$ holds for every $v\in V(G)$, then the subgraph of G with vertex set V(G) and edge set $F_h$, denoted by $G[F_h]$, is called a fractional [a, b]-factor of G with indicator function h, where $E_G(v)$ denotes the set of edges incident with v in G and $F_h=\{e\in E(G):h(e)>0\}$. A graph G is defined as a fractional [a, b]-deleted graph if for any $e\in E(G)$, $G-e$ contains a fractional [a, b]-factor. The size, spectral radius and signless Laplacian spectral radius of G are denoted by e(G), $\rho (G)$ and q(G), respectively. In this paper, we establish a lower bound on the size, spectral radius and signless Laplacian spectral radius of a graph G to guarantee that G is a fractional [a, b]-deleted graph.