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


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.


Graph coloring Planar graphs Guarding problems 


  1. 1.
    Alon, N.: Problems and results in extremal combinatorics, ii. Discrete Math. 308, 4460–4472 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Krech, A., Szabó, T.: Turán’s theorem in the hypercube. SIAM J. Discrete Math. 21, 66–72 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bose, P., Shermer, T., Toussaint, G., Zhu, B.: Guarding polyhedral terrains. Comput. Geom.: Theory Appl. 7(3), 173–185 (1997) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bose, P., Kirkpatrick, D., Li, Z.: Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces. Comput. Geom.: Theory Appl. 26(3), 209–219 (2003) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Dimitrov, D., Horev, E., Krakovski, R.: A note on polychromatic coloring of rectangular partitions. Discrete Math. 30(6), 2957–2960 (2009) CrossRefGoogle Scholar
  6. 6.
    Dinitz, Y., Katz, M.J., Krakovski, R.: Guarding rectangular partitions. In: Abstracts 23rd European Workshop on Computational Geometry, pp. 30–33 (2007) Google Scholar
  7. 7.
    Horev, E., Katz, M.J., Krakovski, R.: Polychromatic coloring of cubic bipartite plane graphs (2009, submitted) Google Scholar
  8. 8.
    Guenin, B.: Packing T-joins and edge colouring in planar graphs. Manuscript (2003) Google Scholar
  9. 9.
    Gupta, R.P.: On the chromatic index and the cover index of a multigraph. In: Theory and Applications of Graphs, Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976. Lecture Notes in Math., vol. 642, pp. 204–215. Springer, Berlin (1978) CrossRefGoogle Scholar
  10. 10.
    Louis Hakimi, S., Kariv, O.: Generalization of edge-coloring in graphs. J. Graph Theory 10, 139–154 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Heawood, P.J.: On the four-color map theorem. Quart. J. Pure Appl. Math. 29, 270–285 (1898) zbMATHGoogle Scholar
  12. 12.
    Hoffmann, F., Kriegel, K.: A graph-coloring result and its consequences for polygon-guarding problems. SIAM J. Discrete Math. 9, 210–224 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Horev, E., Krakovski, R.: Face-respecting colorings of bounded degree plane graphs. Manuscript (2007) Google Scholar
  14. 14.
    Kempe, A.B.: On the geographical problem of four colors. Am. J. Math. 2(3), 193–200 (1879) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Keszegh, B.: Polychromatic colorings of n-dimensional guillotine-partitions In: COCOON, pp. 110–118 (2008) Google Scholar
  16. 16.
    Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986) Google Scholar
  17. 17.
    Mohar, B., Skrekovski, R.: The Grötzsch theorem for the hypergraph of maximal cliques. Electron. J. Comb. 6, R26 (1999) MathSciNetGoogle Scholar
  18. 18.
    Moret, B.M.E.: Planar NAE3SAT is in P. SIGACT News 19(2), 51–54 (1988) CrossRefGoogle Scholar
  19. 19.
    Offner, D.: Polychromatic colorings of subcubes of the hypercube. SIAM J. Discrete Math. 22(2), 450–454 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Seymour, P.D.: On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte. Proc. Lond. Math. Soc. 3, 423–460 (1979) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Steinberg, R.: The state of the three color problem. Ann. Discrete Math. 55, 211–248 (1993) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Stockmeyer, L.: Planar 3-colorability is polynomial complete. SIGACT News 5(3), 19–25 (1973) CrossRefGoogle Scholar
  23. 23.
    West, D.B.: Introduction to Graph Theory. Prentice Hall, New York (1996) zbMATHGoogle Scholar

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

Personalised recommendations