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 \(\cal NP\)-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.
KeywordsPlanar Graph Parallel Composition Simple Cycle Outerplanar Graph Biconnected Graph
Unable to display preview. Download preview PDF.
- 3.Angelini, P., Frati, F.: Acyclically 3-colorable planar graphs. Tech. Report RT-DIA-147-2009, Dept. of Computer Science and Automation, University of Roma Tre. (2009), http://web.dia.uniroma3.it/ricerca/rapporti/rt/2009-147.pdf
- 12.Kostochka, A.V.: Upper Bounds of Chromatic Functions of Graphs. PhD thesis, University of Novosibirsk, in Russian (1978)Google Scholar
- 15.Robertson, N., Sanders, D.P., Seymour, P.D., Thomas, R.: Efficiently four-coloring planar graphs. In: STOC, pp. 571–575 (1996)Google Scholar