A relaxed Hadwiger's Conjecture for list colorings

Authors:
Ken-ichi Kawarabayashi;Bojan Mohar
Affiliations:
National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan;Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

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Abstract

Hadwiger's Conjecture claims that any graph without K"k as a minor is (k-1)-colorable. It has been proved for 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"@?"1"5"."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.