On fractional (g, f, n)-critical graphs

Let G be a graph with vertex set V(G). For any S subset of V(G) we use omega(G - S) to denote the number of components of G - S. The toughness of G, t(G), is defined as t(G) = min{vertical bar S vertical bar/omega(G - S)vertical bar S subset of V(G), omega(G - S) > 1} if G is not complete; otherwise, set t(G) = +infinity. In this paper, we consider the relationship between the toughness and fractional (g, f, n)-critical graphs. It is proved that a graph G is a (g, f, n) -critical graph if t(G) >= (b - 1)(b + n + 1)/a, where a, b, n are integers such that 1 <= a <= b and b >= (1 + root(4n + 5)/2.