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Discrete & Computational Geometry

, Volume 42, Issue 3, pp 421–442 | Cite as

Polychromatic Colorings of Plane Graphs

  • Noga Alon
  • Robert BerkeEmail author
  • Kevin Buchin
  • Maike Buchin
  • Péter Csorba
  • Saswata Shannigrahi
  • Bettina Speckmann
  • Philipp Zumstein
Open Access
Article

Abstract

We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by (3g−5)/4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and 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 k colors in which all colors appear in every face is in ℘ for k=2 and is \(\mathcal{NP}\) -complete for k=3,4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in ℘, and for the others it is \(\mathcal{NP}\) -complete.

Keywords

Graph coloring Planar graphs Guarding problems 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Noga Alon
    • 1
  • Robert Berke
    • 2
    Email author
  • Kevin Buchin
    • 3
  • Maike Buchin
    • 3
  • Péter Csorba
    • 4
  • Saswata Shannigrahi
    • 5
  • Bettina Speckmann
    • 4
  • Philipp Zumstein
    • 6
  1. 1.Tel Aviv UniversityTel AvivIsrael
  2. 2.Tokyo Institute of TechnologyTokyoJapan
  3. 3.Universiteit UtrechtUtrechtThe Netherlands
  4. 4.TU EindhovenEindhovenThe Netherlands
  5. 5.Tata Inst. of Fund. ResearchMumbaiIndia
  6. 6.ETH ZürichZürichSwitzerland

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