A note on the approximability of the toughness of graphs

We show that, if NP≠ZPP, for any ε>0, the toughness of a graph with n vertices is not approximable in polynomial time within a factor of 12 (n/2)1−ε. We give a 4-approximation for graphs with toughness bounded by 13 and we show that this result cannot be generalized to graphs with a bounded toughness. More exactly we prove that there is no constant approximation for graphs with bounded toughness, unless P=NP.