Toughness of Ka,t-Minor-Free Graphs

The toughness of a non-complete graph G is the minimum value of |S| ω(G−S) among all separating vertex sets S ⊂ V (G), where ω(G − S) > 2 is the number of components of G−S. It is well-known that every 3-connected planar graph has toughness greater than 1/2. Related to this property, every 3-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a 2-walk, etc. Realizing that 3-connected planar graphs are essentially the same as 3-connected K3,3-minor-free graphs, we consider a generalization to a-connected Ka,t-minor-free graphs, where 3 6 a 6 t. We prove that there exists a positive constant h(a, t) such that every a-connected Ka,t-minor-free graph G has toughness at least h(a, t). For the case where a = 3 and t is odd, we obtain the best possible value for h(3, t). As a corollary it is proved that every such graph of order n contains a cycle of length Ω(logh(a,t) n).