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Graphs and Combinatorics

, Volume 33, Issue 4, pp 817–832 | Cite as

New Bounds for Facial Nonrepetitive Colouring

  • Prosenjit Bose
  • Vida Dujmović
  • Pat MorinEmail author
  • Lucas Rioux-Maldague
Original Paper

Abstract

We prove that the facial nonrepetitive chromatic number of any outerplanar graph is at most 11 and of any planar graph is at most 22.

References

  1. 1.
    Allouche J.-P., Shallit J.: The ubiquitous Prouhet–Thue–Morse sequence. In: Ding, C., Helleseth, T., Niederreiter, H. (eds.) Sequences and their Applications Proceedings of SETA ‘98 , pp. 1–16. Springer, Berlin (1999). http://www.springer.com/la/book/9781852331962
  2. 2.
    Alon, N., Grytczuk, J., Hałuszczak, M., Riordan, O.: Nonrepetitive colorings of graphs. Random Struct. Algorithms 21(3–4), 336–346 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barát, J., Czap, J.: Facial nonrepetitive vertex coloring of plane graphs. J. Gr. Theory 74(1), 115–121 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barát, J., Varjú, P.P.: On square-free vertex colorings of graphs. Stud. Sci. Math. Hung. 44(3), 411–422 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barát, J., Wood D.R.: Notes on nonrepetitive graph colouring. Electron. J. Comb. 15(1)–99 (2008) (Research Paper) (13)Google Scholar
  6. 6.
    Bondy, J.-A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics. Springer, New York (2007)Google Scholar
  7. 7.
    Brešar, B., Grytczuk, J., Klavžar, S., Niwczyk, S., Peterin, I.: Nonrepetitive colorings of trees. Discret. Math. 307(2), 163–172 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Currie, J.D.: There are ternary circular square-free words of length n for \(n\ge 18\). Electron. J. Comb. 9(1), N10 (2002)zbMATHGoogle Scholar
  9. 9.
    Dujmović, V., Frati, F., Joret, G., Wood, D.R.: Nonrepetitive colourings of planar graphs with \(O(\log n)\) colours. Electron. J. Comb. 20(1), P51 (2013)zbMATHGoogle Scholar
  10. 10.
    Dujmović, V., Joret, G., Kozik, J., Wood, D.R.: Nonrepetitive colouring via entropy compression. Combinatorica (2013) (in press)Google Scholar
  11. 11.
    Fiorenzi, F., Ochem, P., de Mendez, P.O., Zhu, X.: Thue choosability of trees. Discret. Appl. Math. 159(17), 2045–2049 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gagol, A., Joret, G., Kozik, J., Micek, P.: Pathwidth and nonrepetitive list coloring (2016). arXiv:1601.01886
  13. 13.
    Gonçalves, D., Montassier, M., Pinlou, A.: Entropy compression method applied to graph colorings. In: 9th international colloquium on graph theory and combinatorics (2014)Google Scholar
  14. 14.
    Grytczuk, J.: Nonrepetitive colorings of graphs—a survey. Int. J. Math. Math. Sci. 2007 (2007) (Article 74639)Google Scholar
  15. 15.
    Grytczuk J.: Nonrepetitive graph coloring. In: Bondy, A., Fonlupt, J., Fouquet, J.-L., Fournier, J.-C., Ramírez Alfonsín, J.L. (eds.) Graph Theory in Paris Proceedings of a Conference in Memory of Claude Berge, pp. 209–218. Springer, Berlin (2007). http://www.springer.com/us/book/9783764372286
  16. 16.
    Grytczuk, J., Kozik, J., Micek, P.: New approach to nonrepetitive sequences. Random Struct. Algorithms 42(2), 214–225 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Harant, J., Jendrol, S.: Nonrepetitive vertex colorings of graphs. Discret. Math. 312(2), 374–380 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Havet, F., Jendrol, S., Soták, R., Škrabul’áková, E.: Facial non-repetitive edge-coloring of plane graphs. J. Gr. Theory 66(1), 38–48 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kozik, J., Micek, P.: Nonrepetitive choice number of trees. SIAM J. Discret. Math. 27(1), 436–446 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kündgen, A., Pelsmajer, M.J.: Nonrepetitive colorings of graphs of bounded tree-width. Discret. Math. 308(19), 4473–4478 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pezarski, A., Zmarz, M.: Non-repetitive 3-coloring of subdivided graphs. Electron. J. Comb. 16(1), N15 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Przybyło, J.: On the facial Thue choice index via entropy compression. J. Gr. Theory 77(3), 180–189 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schreyer, J., Škrabul’áková, E.: On the facial Thue choice index of plane graphs. Discret. Math. 312(10), 1713–1721 (2012)Google Scholar
  24. 24.
    Schreyer, J., Škrabul’áková, E.: Total Thue colourings of graphs. Eur. J. Math. 1(1), 186–197 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Thue, A.: Über unendliche zeichenreichen. Norske Vid Selsk. Skr. I. Mat. Nat. Kl. Christiana 7, 1–22 (1906)Google Scholar

Copyright information

© Springer Japan 2017

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Vida Dujmović
    • 2
  • Pat Morin
    • 1
    Email author
  • Lucas Rioux-Maldague
    • 3
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  3. 3.GooglePittsburghUSA

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