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Structural Parameterizations of Clique Coloring

Authors Lars Jaffke , Paloma T. Lima, Geevarghese Philip

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Lars Jaffke
  • University of Bergen, Norway
Paloma T. Lima
  • University of Bergen, Norway
Geevarghese Philip
  • Chennai Mathematical Institute, India
  • UMI ReLaX, Chennai, India


The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute.

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Lars Jaffke, Paloma T. Lima, and Geevarghese Philip. Structural Parameterizations of Clique Coloring. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 49:1-49:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an 𝒪^⋆(q^{tw})-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • clique coloring
  • treewidth
  • clique-width
  • structural parameterization
  • Strong Exponential Time Hypothesis


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  1. Thomas Andreae, Martin Schughart, and Zsolt Tuza. Clique-transversal sets of line graphs and complements of line graphs. Discrete Mathematics, 88(1):11-20, 1991. Google Scholar
  2. Gábor Bacsó, Sylvain Gravier, András Gyárfás, Myriam Preissmann, and András Sebo. Coloring the maximal cliques of graphs. SIAM Journal on Discrete Mathematics, 17(3):361-376, 2004. Google Scholar
  3. Gábor Bacsó and Zsolt Tuza. Clique-transversal sets and weak 2-colorings in graphs of small maximum degree. Discrete Mathematics and Theoretical Computer Science, 11(2):15-24, 2009. Google Scholar
  4. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: fast subset convolution. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC 2007), pages 67-74, San Diego, California, USA, June 11-13 2007. ACM. Google Scholar
  5. J. A. Bondy and U. S. R. Murty. Graph Theory. Springer, 2008. Google Scholar
  6. C. N. Campos, Simone Dantas, and Célia Picinin de Mello. Colouring clique-hypergraphs of circulant graphs. Electronic Notes in Discrete Mathematics, 30:189-194, 2008. Google Scholar
  7. Márcia R. Cerioli and André L. Korenchendler. Clique-coloring circular-arc graphs. Electronic Notes in Discrete Mathematics, 35:287-292, 2009. Google Scholar
  8. Pierre Charbit, Irena Penev, Stéphan Thomassé, and Nicolas Trotignon. Perfect graphs of arbitrarily large clique-chromatic number. Journal of Combinatorial Theory, Series B, 116:456-464, 2016. Google Scholar
  9. Maria Chudnovsky and Irene Lo. Decomposing and clique-coloring (diamond, odd-hole)-free graphs. Journal of Graph Theory, 86(1):5-41, 2017. Google Scholar
  10. Manfred Cochefert and Dieter Kratsch. Exact algorithms to clique-colour graphs. In Viliam Geffert, Bart Preneel, Branislav Rovan, Julius Stuller, and A Min Tjoa, editors, Proceedings of the 40th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2014), volume 8327 of LNCS, pages 187-198. Springer, 2014. Google Scholar
  11. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. Google Scholar
  12. 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, 12(3):41:1-41:24, 2016. Google Scholar
  13. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  14. David Défossez. Clique-coloring some classes of odd-hole-free graphs. Journal of Graph Theory, 53(3):233-249, 2006. Google Scholar
  15. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013. Google Scholar
  16. Dwight Duffus, Bill Sands, Norbert Sauer, and Robert E. Woodrow. Two-colouring all two-element maximal antichains. Journal of Combinatorial Theory, Series A, 57(1):109-116, 1991. Google Scholar
  17. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Intractability of clique-width parameterizations. SIAM Journal on Computing, 39(5):1941-1956, 2010. Google Scholar
  18. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Clique-width III: Hamiltonian cycle and the odd case of graph coloring. ACM Transactions on Algorithms, 15(1):9:1-9:27, 2019. Google Scholar
  19. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  20. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. Google Scholar
  21. Lars Jaffke and Bart M. P. Jansen. Fine-grained parameterized complexity analysis of graph coloring problems. In Dimitris Fotakis, Aris Pagourtzis, and Vangelis Th. Paschos, editors, Proceedings of the 10th International Conference on Algorithms and Complexity, (CIAC 2017), volume 10236 of LNCS, pages 345-356, Athens, Greece, May 24-26 2017. Springer. Google Scholar
  22. Lars Jaffke, Paloma T. Lima, and Daniel Lokshtanov. b-Coloring parameterized by clique-width. CoRR, abs/2003.04254, 2020. Google Scholar
  23. Sulamita Klein and Aurora Morgana. On clique-colouring of graphs with few P₄’s. Journal of the Brazilian Computer Society, 18(2):113-119, 2012. Google Scholar
  24. Jan Kratochvíl and Zsolt Tuza. On the complexity of bicoloring clique hypergraphs of graphs. Journal of Algorithms, 45(1):40-54, 2002. Google Scholar
  25. Lásló Lovász. Combinatorial Problems and Exercises. North-Holland Publishing Co., 1993. Google Scholar
  26. Dániel Marx. Complexity of clique coloring and related problems. Theoretical Computer Science, 412(29):3487-3500, 2011. Google Scholar
  27. Bojan Mohar and Riste Skrekovski. The Grötzsch theorem for the hypergraph of maximal cliques. Electronic Journal of Combinatorics, 6(1):128, 1999. Google Scholar
  28. Irena Penev. Perfect graphs with no balanced skew-partition are 2-clique-colorable. Journal of Graph Theory, 81(3):213-235, 2016. Google Scholar
  29. Michaël Rao. Décompositions de graphes et algorithmes efficaces. PhD thesis, University of Metz, 2006. Google Scholar
  30. Michaël Rao. Clique-width of graphs defined by one-vertex extensions. Discrete Mathematics, 308(24):6157-6165, 2008. Google Scholar
  31. Erfang Shan, Zuosong Liang, and Liying Kang. Clique-transversal sets and clique-coloring in planar graphs. European Journal of Combinatorics, 36:367-376, 2014. Google Scholar
  32. Virginia Vassilevska Williams. Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis (invited talk). In Thore Husfeldt and Iyad A. Kanj, editors, Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC 2015), volume 43 of LIPIcs, pages 17-29. Schloss Dagstuhl, 2015. Google Scholar
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