Isolated Toughness and Existence of f-factors

Let G be a graph with vertex set V (G) and edge set E(G). The isolated toughness of G is defined as I(G) = min{|S|/i(G − S) | S ⊆ V (G), i(G−S) ≥ 2} if G is not complete; otherwise, set I(G) = |V (G)|−1. Let f and g be two nonnegative integer-valued functions defined on V (G) satisfying a ≤ g(x) ≤ f(x) ≤ b. The purpose in this paper are to present sufficient conditions in terms of the isolated toughness and the minimum degree for graphs to have f-factors and (g, f)-factors (g < f). If g(x) ≡ a < b ≡ f(x), the conditions can be weakened