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 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 VG) with = n, G - N contains a fractional (g, f)-factor. In this paper, we prove that G is fractional critical if (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 Ŋ(X) is sharp. , journal=Utilitas Mathematica , author=Sun, Zhiren and Zhou, Sizhong, year=2017, month=Mar.