REFINED LIST VERSION OF HADWIGER'S CONJECTURE

Assume lambda = {k(1), k(2),..., k(q)} is a partition of k(lambda) = Sigma(q)(i=1) k(i). A lambda-list assignment of G is a k lambda -list assignment L of G such that the color set U-upsilon is an element of V(G) L(v) can be partitioned into |lambda| = q sets C-1, C-2,. C-q such that for each i and each vertex v of G,|L(v) boolean AND C-i| >= k(i). We say G is lambda-choosable if G is L-colorable for any lambda-list assignment L of G. The concept of lambda-choosability is a refinement of choosability that puts k-choosability and k-colorability in the same framework. If |lambda| is close to k(lambda), then lambda-choosability is close to k(lambda)-colorability; if |lambda| is close to 1, then lambda-choosability is close to k(lambda)-choosability. This paper studies Hadwiger's conjecture in the context of lambda-choosability. Hadwiger's conjecture is equivalent to saying that every K-t -minor-free graph is {1 star(t - 1)}-choosable for any positive integer t, where {1 star (t - 1)} is the multiset consisting of t - 1 copies of 1. We prove that for t >= 5, for any partition lambda of t - 1 other than {1 star (t - 1)}, there is a K-t -minor-free graph G that is not lambda-choosable. We then construct several types of K-t -minor-free graphs that are not lambda-choosable, where k(lambda) - (t - 1) gets larger as k(lambda) - |lambda| gets larger. In particular, for any q and any is an element of > 0, there exists t(0) such that for any t >= t(0), for any partition lambda of [(2 - is an element of)t] with |lambda| = q, there is a K-t -minor-free graph that is not lambda-choosable. The q= 1 case of this result was recently proved by Steiner, and our proof uses a similar argument. We also generalise this result to (a, b) -list coloring.