Isolated Toughness and Fractional (g, f)-Factors of Graphs

Let G be a graph, and let a and b be nonnegative integers such that 1 <= a <= b, and let g and f be two nonnegative integer-valued functions defined on V(G) such that a <= g(x) <= f (x) <= b for each x is an element of V(G). A spanning subgraph F of G is called a fractional (g, f)-factor if g(x) <= d(G)(h)(x) <= f(x) for all x is an element of V(G), where d(G)(h)(x) = Sigma(e is an element of Ex) h(e) is the fractional degree of x is an element of V(F) with E-x = {e : e = xy is an element of E(G)}. The isolated toughness I(G) of a graph G is defined as follows: If G is a complete graph, then I(G) = +infinity; else, I(G) = min{vertical bar S vertical bar/i(G-S) : S subset of V(G), i(G - S) >= 2}, where i(G - S) denotes the number of isolated vertices in G - S. In this paper, we prove that G has a fractional (g, f)-factor if delta(G) >= I(G) >= b(b-1)/a+1. This result is best possible in some sense.