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

Abstract

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 and p ≥ (a+b-1)(a+b+-2)+ bn-1 / (a+r)let g and be two integer-valued functions defined on V(G) such that a ≤ g(x) ≤ f(x) - r ≤ b - r for every x € V(G). A graph G is said to be fractional (g,f,n)-critical if for any N C V{G) with \N\ = n, G - N contains a fractional (g, f)-factor. In this paper, we prove that G is fractional critical if \Na(X)\ for every non-empty independent subset X of V(G), and 6(G) > {b-r-l)p+a+b+bn-2 purthermorei the lower bound qn \NG(X)\ is sharp.