Maximum and minimum toughness of graphs of small genus

A new lower bound on the toughness t(G) of a graph G in terms of its connectivity rc(G) and genus gamma(G) is obtained. For gamma > 0, the bound is sharp via an infinite class of extremal graphs all of girth 4. For planar graphs, the bound is t(G)> kappa(G)/2 - 1. For kappa = 1 this bound is not sharp, but for each kappa = 3,4,5 and any epsilon > 0, infinite families of graphs {G(kappa,epsilon)} are provided with kappa(G(kappa,epsilon)) = kappa, but t(G(kappa,epsilon)) < kappa/2 - 1 + epsilon. Analogous investigations on the torus are carried out, and finally the question of upper bounds is discussed. Several unanswered questions are posed.