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 $g(x)\le f(x)$ 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.