Maximum and minimum toughness of graphs of small genus Original Research Article

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