On Vertex Coloring Without Monochromatic Triangles

  • Michał Karpiński
  • Krzysztof Piecuch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10846)


We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs – the triangle-free chromatic number \(\chi _3\). We bound \(\chi _3\) by other known structural parameters. We also present two classes of graphs with interesting coloring properties, that play pivotal role in proving useful observations about our problem.


  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angelini, P., Frati, F.: Acyclically 3-colorable planar graphs. J. Comb. Optim. 24(2), 116–130 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brooks, R.: On colouring the nodes of a network. Math. Proc. Camb. Philos. Soc. 37(2), 194–197 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Courcelle, B.: The monadic second-order logic for graphs I: recognizable of fiite graphs. Inf. Comput. 85, 12–75 (1990)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dailey, D.P.: Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete. Discret. Math. 30(3), 289–293 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deb R.: An efficient nonparametric test of the collective household model (2008). SSRN:
  7. 7.
    Dinur, I., Regev, O., Smyth, C.: The hardness of 3-uniform hypergraph coloring. In: The 43rd Annual IEEE Symposium on Foundations of Computer Science, pp. 33–40 (2002)Google Scholar
  8. 8.
    Erdos, P.: Graph thoery and probability. Canad. J. Math. 11, 34–38 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fiala, J., Golovach, P., Kratochvíl, J.: Parametrized complexity of coloring problems: treewidth versus vertex cover. Theor. Comput. Sci. 412, 2513–2523 (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Formanowicz, P., Tanaś, K.: A survey of graph coloring - its types, methods and applications. Found. Comput. Decis. Sci. 37(3), 223–238 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jain, P.: On a variant of Monotone NAE-3SAT and the Triangle-Free Cut problem. Pre-print: arXiv:1003.3704 [cs.CC] (2010)
  12. 12.
    Kaiser, T., Škrekovski, R.: Planar graph colorings without short monochromatic cycles. J. Graph Theory 46(1), 25–38 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karpiński, M.: Vertex 2-coloring without monochromatic cycles of fixed size is NP-complete. Theor. Comput. Sci. 659, 88–94 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Karpiński, M., Piecuch, K.: On vertex coloring without monochromatic triangles. Pre-print, arXiv:1710.07132 [cs.DS] (2017)
  15. 15.
    Kawarabayashi, K., Ozeki, K.: A simple algorithm for 4-coloring 3-colorable planar graphs. Theor. Comput. Sci. 411(26–28), 2619–2622 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mycielski, J.: Sur le coloriage des graphs. Colloquium Mathematicae 3(2), 161–162 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schaefer, T.J.: The complexity of satisfiability problems. In: STOC 1978, pp. 216–226 (1978)Google Scholar
  18. 18.
    Shitov, Y.: A tractable NP-completeness proof for the two-coloring without monochromatic cycles of fixed length. Theor. Comput. Sci. 674, 116–118 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Skulrattanakulchai, S.: Delta-list vertex coloring in linear time. Inf. Process. Lett. 98(3), 101–106 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Thomassen, C.: 2-List-coloring planar graphs without monochromatic triangles. J. Comb. Theory Ser. B 98, 1337–1348 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland

Personalised recommendations