Binding number and minimum degree for the existence of (g,f,n)-critical graphs

Let G be a graph of order p. The binding number of G is defined as bind}(G):=min NG(X)|}/{|X|} θ ≠ X ⊂ V(G) and}N G(X) ≠ V(G)}. Let g(x) and f(x) be two nonnegative integer-valued functions defined on V(G) with g(x) ≤ f(x) for any x V(G). A graph G is said to be (g,f,n)-critical if G-N has a (g,f)-factor for each N ⊂ V(G) with |N|=n. If g(x) = a and f(x) = b for all x V(G), then a (g,f,n)-critical graph is an (a,b,n)-critical graph. In this paper, several sufficient conditions on binding number and minimum degree for graphs to be (a,b,n)-critical or (g,f,n)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.