An existence theorem on fractional (g, f, n)-critical graphs

Let a, b, r and n be four nonnegative integers with 1 <= a <= b - r, and let G be a graph of order p with p >= (a+b-1)(a+b-2)+bn-1/a+r, and let g and f be two integer-valued functions defined on V(G) such that a <= g(x) <= f (x) - r <= b - r for every x is an element of V (G). A graph G is said to be fractional (g, f, n)-critical if for any N subset of V (G) with vertical bar N vertical bar = n, G - N contains a fractional (g, f)-factor. In this paper, we prove that G is fractional (g, f,n)-critical if vertical bar N-G(X)vertical bar > (b-r-l)p+vertical bar X vertical bar+bn-1/a+b-1 for every non-empty independent subset X of V (G), and delta(G) > (b-r-1)p+a+b+bn-2/a+b-1 Furthermore, the lower bound on vertical bar NG(X)vertical bar is sharp.