Polychromatic colorings of plane graphs

  • Authors:

  • Noga Alon;Robert Berke;Kevin Buchin;Maike Buchin;Péter Csorba;Saswata Shannigrahi Shannigrahi;Bettina Speckmann;Philipp Zumstein

  • Affiliations:

  • Schools of Mathematics and Computer Science Tel Aviv University, Tel Aviv , Israel;Department of Computer Science ETH Zürich, Zurich, Switzerland;Department of Information and Computing Sciences Universiteit Utrecht, Utrecht, Netherlands;Department of Information and Computing Sciences Universiteit Utrecht, Utrecht, Netherlands;Department of Mathematics and Computer Science TU Eindhoven, Eindhoven, Netherlands;School of Technology and Computer Science Tata Institute of Fundamental Research, Tata, India;Department of Mathematics and Computer Science TU Eindhoven, Eindhoven, Netherlands;Department of Computer Science ETH Zürich, Zürich, Switzerland

  • Venue:
  • Proceedings of the twenty-fourth annual symposium on Computational geometry
  • Year:
  • 2008

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Abstract

We show that the vertices of any plane graph in which every face is of size at least g can be colored by (3g Àý 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of size 3 or 4 only. If all faces are of size 3 this can be decided in polynomial time.