Abstract
In this paper we study the planar graphs that admit an acyclic 3-coloring. We show that testing acyclic 3-colorability is $\cal NP$-hard 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 is acyclically 3-colorable. Finally, we characterize the series-parallel graphs such that every 3-coloring is acyclic and we provide a linear-time recognition algorithm for such graphs.
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