List improper colorings of planar graphs with prescribed girth

A graph G is m-choosable with impropriety d, or simply (m,d)*-choosable, if for every list assignment L, where \ L(upsilon)\ greater than or equal to m for every upsilon is an element of V(G), there exists an L-coloring of G such that every vertex of G has at most d neighbors colored with the same color as itself. Denote by g(d) the smallest number such that every planar graph of girth at least g(d) is (2,d)*-choosable. In this paper it is shown that g(1) less than or equal to 9, g(2) less than or equal to 7, g(3) less than or equal to 6 and g(d) = 5 for every d greater than or equal to 4. (C) 2000 Elsevier Science B.V. All rights reserved.