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

Suppose G is a k-connected graph that does not contain K_k as a minor. What does G look like? This question is motivated by Hadwiger’s conjecture (Vierteljahrsschr. Naturforsch. Ges. Zürich 88 (1943) 133) and a deep result of Robertson and Seymour (J. Combin. Theory Ser. B. 89 (2003) 43).It is easy to see that such a graph cannot contain a (k−1)-clique, but could contain a (k−2)-clique, as K_k−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)).