Isolated Toughness Condition for a Graph to be a Fractional (g, f,n)-Critical Graph

Abstract

Let $i(G)$ be the number of isolated vertices in graph $G$. The isolated toughness of $G$ is defined as $I(G)=+\mathrm{\infty }$ if $G$ is complete; $I(G)=\text{min}\{|S|/i(G-S):S\subseteq V(G),i(G-S)\ge 2\}$ otherwise. In this paper, we determine that $G$ is a fractional $(g,f,n)$-critical graph if $I(G)\ge \frac{b^2+bn–1}{a}$ if $b>a$; $I(G)\ge b+n$ if $a=b$.