Improper choosability of graphs of nonnegative characteristic

A graph G is called (k, d)*-choosable if, for every list assignment L with vertical bar L(v)vertical bar = k for all V E V(G), there is an L-coloring of G such that every vertex has at most d neighbors having the same color as itself. Let G be a graph embeddable in a Surface of nonnegative characteristic. In this paper, we prove: (1) If G contains no k-cycle with a chord for all k = 4, 5. 6, then G is (3, 1)*-choosable; (2) If G contains neither 5-cycle with a chord nor 6-cycle with a chord, then G is (4, 1)*-choosable. (c) 2008 Elsevier Ltd. All rights reserved.