Online Chromatic Number is PSPACE-Complete

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)


In the online graph coloring problem, vertices from a graph G, known in advance, arrive in an online fashion and an algorithm must immediately assign a color to each incoming vertex v so that the revealed graph is properly colored. The exact location of v in the graph G is not known to the algorithm, since it sees only previously colored neighbors of v. The online chromatic number of G is the smallest number of colors such that some online algorithm is able to properly color G for any incoming order. We prove that computing the online chromatic number of a graph is PSPACE-complete.


Incoming Vertex Online Graph Coloring Online Algorithm Binomial Tree Supernode 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank Christian Kudahl and their supervisor Jiří Sgall for useful discussions on the problem.


  1. 1.
    Bean, D.R.: Effective coloration. J. Symbolic Logic 41(2), 469–480 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Böhm, M., Veselý, P.: Online chromatic number is PSPACE-complete, arXiv preprint (2016).
  3. 3.
    Gyárfás, A., Lehel, J.: First fit and on-line chromatic number of families of graphs. Ars Combinatoria 29C, 168–176 (1990)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gyárfás, A., Kiraly, Z., Lehel, J.: On-line graph coloring and finite basis problems. In: Combinatorics: Paul Erdos is Eighty, vol. 1, pp. 207–214 (1993)Google Scholar
  5. 5.
    Halldórsson, M.M.: Parallel and on-line graph coloring. J. Algorithms 23, 265–280 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Halldórsson, M.M.: Online coloring known graphs. Electron. J. Combinatorics 7(1), R7 (2000)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Halldórsson, M.M., Szegedy, M.: Lower bounds for on-line graph coloring. Theor. Comput. Sci. 130(1), 163–174 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kierstad, H.: On-line coloring k-colorable graphs. Israel J. Math. 105, 93–104 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kudahl, C.: On-line graph coloring. Master’s thesis, University of Southern Denmark (2013)Google Scholar
  10. 10.
    Kudahl, C.: Deciding the on-line chromatic number of a graph with pre-coloring is PSPACE-complete. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 313–324. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  11. 11.
    Lovász, L., Saks, M., Trotter, W.T.: An on-line graph coloring algorithm with sublinear performance ratio. Ann. Discrete Math. 43, 319–325 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Computer Science Institute of Charles UniversityPragueCzech Republic

Personalised recommendations