On the connectivity of minimum and 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 ⌈ frac(2 k, 27) ⌉-connected. This improves the previous known bound due to Mader [W. Mader, Über 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 Kk-minor. This problem is still open, but together with the recent result in [T. Böhme, 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 ⌈ frac(k, 3) ⌉-connected. This is also the first result on the vertex connectivity of minimum counterexamples to Hadwiger's Conjecture for general k. © 2006 Elsevier Inc. All rights reserved.

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Kawarabayashi, K. ichi. (2007). On the connectivity of minimum and minimal counterexamples to Hadwiger’s Conjecture. Journal of Combinatorial Theory. Series B, 97(1), 144–150. https://doi.org/10.1016/j.jctb.2006.04.004

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