Tight toughness condition for fractional \((g,f,n)\)-critical graphs (Q2870474)

Assume that \(f\) and \(g\) are two integer-valued functions defined on the vertex set of a graph \(G\) such that \(0\leq g(x)\leq f(x)\) for each vertex \(x\). A spanning subgraph \(F\) of \(G\) is called a \((g,f)\)-factor if the degree of \(x\) in \(F\) is between \(g(x)\) and \(f(x)\), for any vertex \(x\) of \(G\). A fractional \((g,f)\)-factor is a function \(h\) defined on the edges of \(G\) such that \(h(e)\in [0,1]\) for each edge \(e\) and \(g(x)\leq \sum_{x\in e}h(x)\leq f(x)\), for any vertex \(x\) of \(G\). A graph is called \((g,f,n)\)-critical if, after deleting any \(n\) vertices from \(G\), the resulting graph has a fractional \((g,f)\)-factor. The toughness of a graph \(G\) is the minimum of \(|S|/c(G\setminus S)\), where the minimum is taken over all disconnecting sets of vertices \(S\) of \(G\) and \(c(G\setminus S)\) denotes the number of components of \(G\setminus S\). In this paper, the authors determine a sufficient condition (in terms of toughness) that implies that a graph is fractional \((g,f,n)\)-critical.