Planar graphs without 3-,7-, and 8-cycles are 3-choosable ∗

A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem states that every planar triangle-free graph is 3-colorable. However, Voigt [13] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangle-free graph without 7- and 8-cycles is 3-choosable.