On \((g,f,n)\)-Critical Graphs

Abstract

Let \(G\) be a graph, and let \(g\) and \(f\) be two integer-valued functions defined on \(V(G)\) such that \(g(x) \leq f(x)\) for all \(x \in V(G)\). A graph \(G\) is called a \((g, f, n)\)-critical graph if \(G-N\) has a \((g, f)\)-factor for each \(N \subseteq V(G)\) with \(|N| = n\). In this paper, a necessary and sufficient condition for a graph to be \((g, f, n)\)-critical is given. Furthermore, the properties of \((g, f, n)\)-critical graphs are studied.