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

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] found such a graph which is not 3-choosable. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4or 5-cycle in exactly one edge. This strengthens the Thomassen’s result [8] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6and 7-cycles is 3-choosable.