Document Open Access Logo

Breaking the 2ⁿ Barrier for 5-Coloring and 6-Coloring

Author Or Zamir

Thumbnail PDF


  • Filesize: 0.94 MB
  • 20 pages

Document Identifiers

Author Details

Or Zamir
  • Blavatnik School of Computer Science, Tel Aviv University, Israel


The author would like to deeply thank Noga Alon for important discussions and insights regarding the subset removal lemma, and Haim Kaplan and Uri Zwick for many helpful discussions and comments on the paper. The author would also like to thank anonymous reviewers for helpful comments.

Cite AsGet BibTex

Or Zamir. Breaking the 2ⁿ Barrier for 5-Coloring and 6-Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 113:1-113:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O^*(2ⁿ) time, as shown by Björklund, Husfeldt and Koivisto in 2009. For k = 3,4, better algorithms are known for the k-coloring problem. 3-coloring can be solved in O(1.33ⁿ) time (Beigel and Eppstein, 2005) and 4-coloring can be solved in O(1.73ⁿ) time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k > 4 no improvements over the general O^*(2ⁿ) are known. We show that both 5-coloring and 6-coloring can also be solved in O((2-ε) ⁿ) time for some ε > 0. As a crucial step, we obtain an exponential improvement for computing the chromatic number of a very large family of graphs. In particular, for any constants Δ,α > 0, the chromatic number of graphs with at least α⋅ n vertices of degree at most Δ can be computed in O((2-ε) ⁿ) time, for some ε = ε_{Δ,α} > 0. This statement generalizes previous results for bounded-degree graphs (Björklund, Husfeldt, Kaski, and Koivisto, 2010) and graphs with bounded average degree (Golovnev, Kulikov and Mihajlin, 2016). We generalize the aforementioned statement to List Coloring, for which no previous improvements are known even for the case of bounded-degree graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Graph algorithms analysis
  • Algorithms
  • Graph Algorithms
  • Graph Coloring


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Richard Beigel and David Eppstein. 3-coloring in time O(1.3289ⁿ). Journal of Algorithms, 54(2):168-204, 2005. Google Scholar
  2. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Trimmed moebius inversion and graphs of bounded degree. Theory of Computing Systems, 47(3):637-654, 2010. Google Scholar
  3. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM Journal on Computing, 39(2):546-563, 2009. Google Scholar
  4. Jesper Makholm Byskov. Enumerating maximal independent sets with applications to graph colouring. Operations Research Letters, 32(6):547-556, 2004. Google Scholar
  5. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. The complexity of satisfiability of small depth circuits. In International Workshop on Parameterized and Exact Computation, pages 75-85. Springer, 2009. Google Scholar
  6. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On problems as hard as CNF-SAT. ACM Transactions on Algorithms (TALG), 12(3):1-24, 2016. Google Scholar
  7. Marek Cygan and Marcin Pilipczuk. Faster exponential-time algorithms in graphs of bounded average degree. Information and Computation, 243:75-85, 2015. Google Scholar
  8. David Eppstein. Small maximal independent sets and faster exact graph coloring. In Workshop on Algorithms and Data Structures, pages 462-470. Springer, 2001. Google Scholar
  9. Fedor V Fomin, Serge Gaspers, and Saket Saurabh. Improved exact algorithms for counting 3-and 4-colorings. In International Computing and Combinatorics Conference, pages 65-74. Springer, 2007. Google Scholar
  10. Fedor V Fomin and Petteri Kaski. Exact exponential algorithms. Communications of the ACM, 56(3):80-88, 2013. Google Scholar
  11. F.V. Fomin and D. Kratsch. Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer Berlin Heidelberg, 2010. Google Scholar
  12. Alexander Golovnev, Alexander S Kulikov, and Ivan Mihajlin. Families with infants: speeding up algorithms for np-hard problems using fft. ACM Transactions on Algorithms (TALG), 12(3):1-17, 2016. Google Scholar
  13. Thomas Dueholm Hansen, Haim Kaplan, Or Zamir, and Uri Zwick. Faster k-SAT algorithms using biased-PPSZ. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 578-589, 2019. Google Scholar
  14. Thore Husfeldt. Graph colouring algorithms, page 277–303. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2015. URL:
  15. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  16. Lars Jaffke and Bart MP Jansen. Fine-grained parameterized complexity analysis of graph coloring problems. In International Conference on Algorithms and Complexity, pages 345-356. Springer, 2017. Google Scholar
  17. Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972. Google Scholar
  18. Robert Kennes. Computational aspects of the mobius transformation of graphs. IEEE Transactions on Systems, Man, and Cybernetics, 22(2):201-223, 1992. Google Scholar
  19. Donald Ervin Knuth. Seminumerical algorithms. The art of computer programming, 2, 1997. Google Scholar
  20. Vipin Kumar. Algorithms for constraint-satisfaction problems: A survey. AI magazine, 13(1):32-32, 1992. Google Scholar
  21. Eugene L Lawler. A note on the complexity of the chromatic number problem, 1976. Google Scholar
  22. László Lovász. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conference of Combinatorics, Graph Theory, and Computing, pages 3-12. Utilitas Mathematica Publishing, 1973. Google Scholar
  23. Burkhard Monien and Ewald Speckenmeyer. Solving satisfiability in less than 2n steps. Discrete Applied Mathematics, 10(3):287-295, 1985. Google Scholar
  24. John W Moon and Leo Moser. On cliques in graphs. Israel journal of Mathematics, 3(1):23-28, 1965. Google Scholar
  25. Ramamohan Paturi, Pavel Pudlák, Michael E Saks, and Francis Zane. An improved exponential-time algorithm for k-SAT. Journal of the ACM (JACM), 52(3):337-364, 2005. Google Scholar
  26. Marvin C Paull and Stephen H Unger. Minimizing the number of states in incompletely specified sequential switching functions. IRE Transactions on Electronic Computers, pages 356-367, 1959. Google Scholar
  27. Gian-Carlo Rota. On the foundations of combinatorial theory i. theory of möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 2(4):340-368, 1964. Google Scholar
  28. Ingo Schiermeyer. Deciding 3-colourability in less than O(1.415ⁿ) steps. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 177-188. Springer, 1993. Google Scholar
  29. T Schoning. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 410-414. IEEE, 1999. Google Scholar
  30. Larry Stockmeyer. Planar 3-colorability is polynomial complete. ACM Sigact News, 5(3):19-25, 1973. Google Scholar
  31. Gerhard J Woeginger. Exact algorithms for NP-hard problems: A survey. In Combinatorial optimization - eureka, you shrink!, pages 185-207. Springer, 2003. Google Scholar
  32. Frank Yates. The design and analysis of factorial experiments. Imperial Bureau of Soil Science Harpenden, UK, 1937. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail