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

Authors:
Ken-ichi Kawarabayashi;Rong Luo;Jianbing Niu;Cun-Quan Zhang
Affiliations:
Graduate School of Information Sciences (GSIS), Tohoku University, Aramaki aza Aoba 09, Aoba-ku Sendai, Miyagi 980-8579, Japan;Department of Mathematics, West Virginia University, Morgantown, WV;Department of Mathematics, West Virginia University, Morgantown, WV;Department of Mathematics, West Virginia University, Morgantown, WV
Venue:
European Journal of Combinatorics - Special issue: Topological graph theory II
Year:
2005

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Abstract

Suppose G is a k-connected graph that does not contain Kk 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 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)).