Connectivities for k-knitted graphs and for minimal counterexamples to Hadwigerʼs Conjecture

For a given subset S⊆V(G) of a graph G, the pair (G,S) is knitted if for every partition of S into non-empty subsets S1,S2,…,St, there are disjoint connected subgraphs C1,C2,…,Ct in G so that Si⊆Ci. A graph G is ℓ-knitted if (G,S) is knitted for all S⊆V(G) with |S|=ℓ. In this paper, we prove that every 9ℓ-connected graph is ℓ-knitted.Hadwigerʼs Conjecture states that every k-chromatic graph contains a Kk-minor. We use the above result to prove that the connectivity of minimal counterexamples to Hadwigerʼs Conjecture is at least k/9, which was proved to be at least 2k/27 in Kawarabayashi (2007) [4].