A note on list improper coloring planar graphs

A graph G is called (k, d)*-choosable if, for every list assignment L satisfying \L(v)\ = k for all v is an element of V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every planar graph without 4-cycles and l-cycles for some l is an element of {5, 6, 7} is (3, 1)*-choosable. (C) 2001 Elsevier Science Ltd. All rights reserved.