Acyclically 3-Colorable Planar Graphs

In this paper we study the planar graphs that admit an acyclic 3-coloring. We show that testing acyclic 3-colorability is NP-hard, even for planar graphs of maximum degree 4, and we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Further, we show that every planar graph has a subdivision with one vertex per edge that admits an acyclic 3-coloring. Finally, we show that every series-parallel graph admits an acyclic 3-coloring and we give a linear-time algorithm for recognizing whether every 3-coloring of a series-parallel graph is acyclic.