On the structure of k-connected graphs without Kk-minor

It is easy to see that such a graph cannot contain a (k−1)-clique, but could contain a (k−2)-clique, as Kk−5+G′, where G′ is a 5-connected planar graph, shows. In this paper, however, we will prove that such a graph cannot contain three “nearly” disjoint (k−2)-cliques. This theorem generalizes some early results by Robertson et al. (Combinatorica 13 (1993) 279) and Kawarabayashi and Toft (Combinatorica (in press)).