3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles

Abstract

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4- or 5-cycle in exactly one edge. This strengthens Thomassen's result [C. Thomassen, J. Combin. Theory Ser. B, 64 (1995), pp. 101–107] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6- and 7-cycles is 3-choosable.