IMPROPER CHOOSABILITY OF PLANAR GRAPHS WITHOUT 4-CYCLES

Let k >= 2 be a positive integer, and let d be a nonnegative integer. A graph G is d-improperly k-choosable, or simply, (k, d)-choosable if, for every list assignment L with vertical bar L(v)vertical bar >= k for every nu is an element of V (G), there exists an L-coloring of G such that each vertex of G has at most d neighbors colored the same color as itself. It is known that every planar graph with cycles of length neither 4 nor k for some k is an element of {5, 6, 7, 8, 9} is (3, 1)-choosable. In this paper, we prove that every planar graph without cycles of length 4 is (3, 1)-choosable. This is best possible in the following two senses: (i) there are planar graphs which are not (3, 1)-colorable, hence not (3, 1)-choosable; (ii) there are planar graphs without 4-cycles which are not (3, 0)-colorable, hence not (3, 0)-choosable.