On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture

Authors:
Ken-ichi Kawarabayashi
Affiliations:
Graduate School of Information Sciences (GSIS), Tohoku University, Aramaki aza Aoba 09, Aoba-ku Sendai-city, Miyagi-pref. 980-8579, Japan
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

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Connectivities for k-knitted graphs and for minimal counterexamples to Hadwiger's Conjecture

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Abstract

The main result of this paper is the following: Any minimal counterexample to Hadwiger's Conjecture for the k-chromatic case is @?2k27@?-connected. This improves the previous known bound due to Mader [W. Mader, Uber trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243-252], which says that any minimal counterexample to Hadwiger's Conjecture for the k-chromatic case is 7-connected for k=7. This is the first result on the vertex connectivity of minimal counterexamples to Hadwiger's Conjecture for general k. Consider the following problem: There exists a constant c such that any ck-chromatic graph has a K"k-minor. This problem is still open, but together with the recent result in [T. Bohme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces large complete bipartite graph minors, preprint], our main result implies that there are only finitely many minimal counterexamples to the above problem when c=27. This would be the first step to attach the above problem. We also prove that the vertex connectivity of minimum counterexamples to Hadwiger's Conjecture is at least @?k3@?-connected. This is also the first result on the vertex connectivity of minimum counterexamples to Hadwiger's Conjecture for general k.