Some Sufficient Conditions for Graphs to Be (g,f,n)-Critical Graphs

Let G be a graph of order p, and let a and b and n be nonnegative integers with 1'a'b, and let g and f be two integer-valued functions defined on V(G) such that a'g(x)'f(x)'b for all x+AFs-V(G). A (g,f)-factor of graph G is defined as a spanning subgraph F of G such that g(x)'dF(x)'f(x) for each x+AFs-V(G). Then a graph G is called a (g,f,n)-critical graph if after deleting any n vertices of G the remaining graph of G has a (g,f)-factor. In this paper, we prove that every graph G is a (g,f,n)-critical graph if its minimum degree is greater than p+a+Y-2'. Furthermore, it is showed that the result in this paper is best possible in some sense.