ALL FRACTIONAL (g, f) -FACTORS IN GRAPHS

Let G a graph, and g, f : V(G) -> N be two functions with g(x) <= f(x) for each vertex x in G. We say that G has all fractional (g, f)-factors if G includes a fractional r-factor for every r : V(G) -> N with g(x) <= r(x) <= f(x) for each vertex x in G. Let H be a subgraph of G. We say that G admits all fractional (g, f) -factors including H if for every r : V(G) -> N with g(x) <= r(x) <= f(x) for each vertex x in G, G includes a fractional r -factor F-h with h(e) = 1 for any e is an element of E(H) , where h: E(G) -> [0,1] is the indicator function of F-h. In this paper, we obtain a characterization for the existence of all fractional (g, f)-factors including H and pose a sufficient condition for a graph to have all fractional (g, f)-factors including H.