A relaxed Hadwigerʹs Conjecture for list colorings

Hadwigerʹs Conjecture claims that any graph without K k as a minor is ( k − 1 ) -colorable. It has been proved for k ⩽ 6 , and is still open for every k ⩾ 7 . It is not even known if there exists an absolute constant c such that any ck-chromatic graph has K k as a minor. Motivated by this problem, we show that there exists a computable constant f ( k ) such that any graph G without K k as a minor admits a vertex partition V 1 , … , V ⌈ 15.5 k ⌉ such that each component in the subgraph induced on V i ( i ⩾ 1 ) has at most f ( k ) vertices. This result is also extended to list colorings for which we allow monochromatic components of order at most f ( k ) . When f ( k ) = 1 , this is a coloring of G. Hence this is a relaxation of coloring and this is the first result in this direction.