Algorithmica

, Volume 58, Issue 3, pp 770–789 | Cite as

Fast 3-coloring Triangle-Free Planar Graphs

Article

Abstract

Although deciding whether the vertices of a planar graph can be colored with three colors is NP-hard, the widely known Grötzsch’s theorem states that every triangle-free planar graph is 3-colorable. We show the first o(n2) algorithm for 3-coloring vertices of triangle-free planar graphs. The time complexity of the algorithm is \(\mathcal{O}(n\log n)\) .

Keywords

Graph algorithms Triangle-free planar graphs Grötzsch’s theorem Coloring Efficient algorithm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kowalik, Ł.: Fast 3-coloring triangle-free planar graphs. In: Albers, S., Radzik, T. (eds.) Proc. 12th Annual European Symposium on Algorithms (ESA 2004). Lecture Notes in Computer Science, vol. 3221, pp. 436–447. Springer, Berlin (2004) Google Scholar
  2. 2.
    Robertson, N., Sanders, D.P., Seymour, P., Thomas, R.: Efficiently four-coloring planar graphs. In: Proc. 28th Symposium on Theory of Computing, pp. 571–575. ACM, New York (1996) Google Scholar
  3. 3.
    Chiba, N., Nishizeki, T., Saito, N.: A linear algorithm for five-coloring a planar graph. J. Algorithms 2, 317–327 (1981) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Garey, M.R., Johnson, D.S.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Grötzsch, H.: Ein dreifarbensatz fur dreikreisfreie netze auf der kuzel. Technical report, Wiss. Z. Martin Luther Univ. Halle Wittenberg, Math.-Nat. Reihe 8 (1959) Google Scholar
  6. 6.
    Thomassen, C.: Grötzsch’s 3-color theorem and its counterparts for the torus and the projective plane. J. Comb. Theory, Ser. B 62, 268–279 (1994) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Thomassen, C.: A short list color proof of Grötzsch’s theorem. J. Comb. Theory, Ser. B 88, 189–192 (2003) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kowalik, Ł., Kurowski, M.: Oracles for bounded-length shortest paths in planar graphs. ACM Trans. Algorithms 2(3), 335–363 (2006) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Borodin, O.V., Raspaud, A.: A sufficient condition for planar graphs to be 3-colorable. J. Comb. Theory, Ser. B 88, 17–27 (2003) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Borodin, O.V., Glebov, A.N., Raspaud, A., Salavatipour, M.R.: Planar graphs without cycles of length from 4 to 7 are 3-colorable. J. Comb. Theory, Ser. B 93(2), 303–311 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dvorak, Z., Král, D., Thomas, R.: Coloring triangle-free graphs on surfaces. In: Algorithms and Computation, 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17–19, 2007, Proceedings. LNCS, vol. 4835, pp. 2–4. Springer, Berlin (2007) Google Scholar
  12. 12.
    West, D.: Introduction to Graph Theory. Prentice Hall, New York (1996) MATHGoogle Scholar
  13. 13.
    Gimbel, J., Thomassen, C.: Coloring graphs with fixed genus and girth. Trans. Am. Math. Soc. 349(11), 4555–4564 (1997) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001) MATHGoogle Scholar
  15. 15.
    Brodal, G.S., Fagerberg, R.: Dynamic representations of sparse graphs. In: Proc. 6th Int. Workshop on Algorithms and Data Structures. LNCS, vol. 1663, pp. 342–351. Springer, Berlin (1999) CrossRefGoogle Scholar
  16. 16.
    Dvořák, Z., Kawarabayashi, K., Thomas, R.: Three-coloring triangle-free planar graphs in linear time. In: SODA’09: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1176–1182 (2009) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of WarsawWarsawPoland

Personalised recommendations