Isolated Toughness and Fractional \((g, f)\)-Factors of Graphs

Abstract

Let \(G\) be a graph, and let \(a\) and \(b\) be nonnegative integers such that \(1 \leq a \leq b\). Let \(g\) and \(f\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\) for each \(x \in V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \((g, f)\)-factor if \(g(x) \leq d_G^h(x) \leq f(x)\) for all \(x \in V(G)\), where \(d_G^h(x) = \sum_{e \in E_x} h(e)\) is the fractional degree of \(x \in V(F)\) with \(E_x = \{e : e = xy \in E(G)\}\). The isolated toughness \(I(G)\) of a graph \(G\) is defined as follows: If \(G\) is a complete graph, then \(I(G) = +\infty\); else, \(I(G) = \min\{ \frac{|S|}{i(G-S)} : S \subseteq V(G), i(G – S) \geq 2 \}\), where \(i(G – S)\) denotes the number of isolated vertices in \(G – S\). In this paper, we prove that \(G\) has a fractional \((g, f)\)-factor if \(\delta(G) \geq I(G) \geq \frac{b(b-1)}{a}+1\). This result is best possible in some sense.