Skip to main content

Online Coloring a Token Graph


We study a combinatorial coloring game between two players, Spoiler and Painter, who alternate turns. First, Spoiler places a new token at a vertex in G, and Painter responds by assigning a color to the new token. Painter must ensure that tokens on the same or adjacent vertices receive distinct colors. Spoiler must ensure that the token graph (in which two tokens are adjacent if and only if their distance in G is at most 1) has chromatic number at most w. Painter wants to minimize the number of colors used, and Spoiler wants to force as many colors as possible. Let \(f(w,G)\) be the minimum number of colors needed in an optimal Painter strategy. The game is motivated by a natural online coloring problem on the real line which remains open. A graph G is token-perfect if \(f(w,G) = w\) for each w. We show that a graph is token-perfect if and only if it can be obtained from a bipartite graph by cloning vertices. We also give a forbidden induced subgraph characterization of the class of token-perfect graphs, which may be of independent interest. When G is not token-perfect, determining \(f(w,G)\) seems challenging; we establish \(f(w,G)\) asymptotically for some of the minimal graphs that are not token-perfect.

This is a preview of subscription content, access via your institution.


  1. Bartal, Y., Fiat, A., Leonardi, S.: Lower bounds for on-line graph problems with application to on-line circuit and optical routing. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, pp. 531–540. ACM, New York (1996)

  2. Bartal, Y., Fiat, A., Leonardi, S.: Lower bounds for on-line graph problems with application to on-line circuit and optical routing. SIAM J. Comput. 36(2), 354–393 (2006)

    Article  MathSciNet  Google Scholar 

  3. Bosek, B., Felsner, S., Kloch, K., Krawczyk, T., Matecki, G., Micek, P.: On-line chain partitions of orders: a survey. Order 29(1), 49–73 (2012)

    Article  MathSciNet  Google Scholar 

  4. Chan, T.M., Zarrabi-Zadeh, H.: A randomized algorithm for online unit clustering. Theory Comput. Syst. 45(3), 486–496 (2009)

    Article  MathSciNet  Google Scholar 

  5. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. (2) 164(1), 51–229 (2006)

    Article  MathSciNet  Google Scholar 

  6. Corneil, D.G., Lerchs, H., Stewart Burlingham, L.: Complement reducible graphs. Discrete Appl. Math. 3(3), 163–174 (1981)

    Article  MathSciNet  Google Scholar 

  7. Ehmsen, M.R., Larsen, K.S.: Better bounds on online unit clustering. Theor. Comput. Sci. 500(Supplement C), 1–24 (2013)

    Article  MathSciNet  Google Scholar 

  8. Epstein, L., Levy, M.: Online interval coloring and variants. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) Automata, Languages and Programming, pp. 602–613. Springer, Berlin (2005)

    Chapter  Google Scholar 

  9. Epstein, L., Van Stee, R.: On the online unit clustering problem. ACM Trans. Algorithms 7(1), 7:1–7:18 (2010)

    Article  MathSciNet  Google Scholar 

  10. Halldórsson, M.M.: Online coloring known graphs. Electron. J. Combin. 7, Research Paper 7, 9 (2000)

  11. Halldórsson, M.M., Szegedy, M.: Lower bounds for on-line graph coloring. Theor. Comput. Sci. 130(1), 163–174 (1994)

    Article  MathSciNet  Google Scholar 

  12. Hougardy, S.: Classes of perfect graphs. Discrete Math. 306(19–20), 2529–2571 (2006)

    Article  MathSciNet  Google Scholar 

  13. Junosza-Szaniawski, K., Rzążewski, P., Sokół, J., Węsek, K.: Online coloring and \(L(2,1)\)-labeling of unit disk intersection graphs. SIAM J. Discrete Math. 32(2), 1335–1350 (2018)

    Article  MathSciNet  Google Scholar 

  14. Kawahara, J., Kobayashi, K.M.: An improved lower bound for one-dimensional online unit clustering. Theor. Comput. Sci. 600(Supplement C), 171–173 (2015)

    Article  MathSciNet  Google Scholar 

  15. Kierstead, H.A.: On-line coloring \(k\)-colorable graphs. Isr. J. Math. 105(1), 93–104 (1998)

    Article  MathSciNet  Google Scholar 

  16. Kierstead, H.A.: Coloring graphs on-line. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms: The State of the Art, pp. 281–305. Springer, Berlin (1998)

    Chapter  Google Scholar 

  17. Lovász, L., Saks, M., Trotter, W.T.: An on-line graph coloring algorithm with sublinear performance ratio. Discrete Math. 75(1–3), 319–325 (1989)

    MathSciNet  MATH  Google Scholar 

  18. McDiarmid, C., Reed, B.: Channel assignment and weighted coloring. Networks 36(2), 114–117 (2000)

    Article  MathSciNet  Google Scholar 

  19. Stahl, S.: n-tuple colorings and associated graphs. J. Combin. Theory Ser. B 20(2), 185–203 (1976)

    Article  MathSciNet  Google Scholar 

Download references


We thank Csaba Biro for preliminary discussions on the value of the game on powers of infinite paths.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Kevin G. Milans.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Milans, K.G., Wigal, M.C. Online Coloring a Token Graph. Graphs and Combinatorics 36, 153–165 (2020).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


  • Combinatorial game
  • Online coloring
  • Graph coloring