Some Existence Theorems on All Fractional (g, f)-factors with Prescribed Properties

Let G be a graph, and g, f: V (G) -> Z (+) with g(x) ae f(x) for each x a V (G). We say that G admits all fractional (g, f)-factors if G contains an fractional r-factor for every r: V (G) -> Z (+) with g(x) ae r(x) ae f(x) for any x a V (G). Let H be a subgraph of G. We say that G has all fractional (g, f)-factors excluding H if for every r: V (G) -> Z (+) with g(x) ae r(x) ae f(x) for all x a V (G), G has a fractional r-factor F (h) such that E(H) a (c) E(F (h) ) = theta, where h: E(G) -> [0, 1] is a function. In this paper, we show a characterization for the existence of all fractional (g, f)-factors excluding H and obtain two sufficient conditions for a graph to have all fractional (g, f)-factors excluding H.