Defective choosability of graphs in surfaces

It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ǫ, and k and d are positive integers such that k > 3 and d is sufficiently large in terms of k and ǫ, then G is (k, d)∗-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the known lower bound on d that suffices for this is reduced, and an analogous result is proved for list colorings (choosability). Also, the recent result of Cushing and Kierstead, that every planar graph is (4, 1)∗-choosable, is extended to K3,3-minor-free and K5-minor-free graphs.