A Neighborhood Condition For Graphs To Have Restricted Fractional (G, F)-Factors

Let h be a function defined on E(G) with h(e) is an element of [0, 1] for any e is an element of E(G). Set d(G)(h)(x) = Sigma(e(sic)x) h(e). If g(x) <= d(G)(h)(x) <= f(x) for every x is an element of V(G), then we call the graph F-h with vertex set V(G) and edge set E-h a fractional (g, f)-factor of G with indicator function h, where E-h = {e : e is an element of E(G), h(e) > 0}. Let M and N be two sets of independent edges of G with M boolean AND N = empty set, vertical bar M vertical bar = m and vertical bar N vertical bar = n. If G admits a fractional (g, f)-factor F-h such that h(e) = 1 for any e is an element of M and h(e) = 0 for any e is an element of N, then we say that G has a fractional (g, f)-factor with the property E(m, n). In this paper, we present a neighborhood condition for the existence of a fractional (g, f)-factor with the property E(1, n) in a graph. Furthermore, it is shown that the neighborhood condition is sharp.