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

Abstract

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). The isolated toughness of \(G\) is defined as
\(I(G) = \min\left\{\frac{|S|}{i(G-S)}: S \subseteq V(G), i(G-S) \geq 2\right\}\)
if \(G\) is not complete. Otherwise, set \(I(G) = |V(G)| – 1\). Let \(a\) and \(b\) be positive integers such that \(1 \leq a \leq b\), and let \(g(x)\) and \(f(x)\) be positive integral-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\). Let \(h(e) \in [0,1]\) be a function defined on \(E(G)\), and let \(d(x) = \sum_{e \in E_x} h(e)\) where \(E_x = \{xy : y \in V(G)\}\). Then \(d(x)\) is called the fractional degree of \(x\) in \(G\). We call \(h\) an indicator function if \(g(x) \leq d(x) \leq f(x)\) holds for each \(x \in V(G)\). Let \(E^h = \{e : e \in E(G), h(e) \neq 0\}\) and let \(G_h\) be a spanning subgraph of \(G\) such that \(E(G_h) = E^h\). We call \(G_h\) a fractional \((g,f)\)-factor. The main results in this paper are to present some sufficient conditions about isolated toughness for the existence of fractional \((g,f)\)-factors. If \(1 = g(x) < f(x) = b\), this condition can be improved and the improved bound is not only sharp but also a necessary and sufficient condition for a graph to have a fractional \([1,b]\)-factor.