Journal of Combinatorial Theory Series B
For a given subset S subset of V (G) of a graph G, the pair (G, S) is knitted if for every partition of S into non-empty subsets S-1, S-2, ... , S-t, there are disjoint connected subgraphs C-1, C-2, ... , C-t in G so that S-i subset of C-i. A graph G is l-knitted if (G, S) is knitted for all S subset of V(G) with vertical bar S vertical bar = l. In this paper, we prove that every 9l-connected graph is l-knitted. Hadwiger's Conjecture states that every k-chromatic graph contains a K-k-minor. We use the above result to prove that the connectivity of minimal counterexamples to Hadwiger's Conjecture is at least k/9, which was proved to be at least 2k/27 in Kawarabayashi (2007) . (C) 2013 Elsevier Inc. All rights reserved.
Kawarabayashi, K. I., & Yu, G. (2013). Connectivities for k-knitted graphs and for minimal counterexamples to Hadwigerʼs Conjecture. Journal of Combinatorial Theory, Series B, 103(3), 320-326.