A Note on Fractional \((g, f, m)\)-Deleted Graphs

Abstract

A graph \(G\) is called a fractional \((g, f, m)\)-deleted graph if after deleting any \(m\) edges, then the resulting graph admits a fractional \((g, f)\)-factor. In this paper, we prove that if \(G\) is a graph of order \(n\), and if \(1 \leq g(x) \leq f(x) \leq 6\) for any \(x \in V(G)\), \(\delta(G) \geq \frac{b^2(i-1)}{a} ++2m\), \(n > \frac{(a+b)(i(a+b)+2m-2)}{a}\) and \(|N_G(x_1) \cup N_G(x_2) \cup \cdots \cup N_G(x_i)| \geq \frac{bn}{a+b} \), for any independent set \(\{x_1, x_2, \ldots,x_i\}\) of \(V(G)\), where \(i \geq 2\), then \(G\) is a fractional \((g, f, m)\)-deleted graph. The result is tight on the neighborhood union condition.