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 is an element of V(G). A (g, f)-factor of graph G is defined as a spanning subgraph F of G such that g(x) <= d(F)(x) <= f(x) for each x is an element of 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 + b - 2 root(a + 1)p - bn + 1. Furthermore, it is showed that the result in this paper is best possible in some sense.