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

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