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Complexity of C_k-Coloring in Hereditary Classes of Graphs

Authors Maria Chudnovsky, Shenwei Huang, Paweł Rzążewski, Sophie Spirkl, Mingxian Zhong



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Author Details

Maria Chudnovsky
  • Princeton University, Princeton, NJ 08544, USA
Shenwei Huang
  • College of Computer Science, Nankai University, Tianjin 300350, China
Paweł Rzążewski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland
Sophie Spirkl
  • Rutgers University, Piscataway, NJ 08854, USA
Mingxian Zhong
  • Lehman College, CUNY, Bronx, NY 10468, USA

Acknowledgements

We are grateful to anonymous reviewers for their comments that helped improve the presentation of the paper.

Cite AsGet BibTex

Maria Chudnovsky, Shenwei Huang, Paweł Rzążewski, Sophie Spirkl, and Mingxian Zhong. Complexity of C_k-Coloring in Hereditary Classes of Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 31:1-31:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.31

Abstract

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs. We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • homomorphism
  • hereditary class
  • computational complexity
  • forbidden induced subgraph

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References

  1. Vladimir E Alekseev. The effect of local constraints on the complexity of determination of the graph independence number. Combinatorial-algebraic methods in applied mathematics, pages 3-13, 1982. Google Scholar
  2. Gábor Bacsó, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Zsolt Tuza, and Erik Jan van Leeuwen. Subexponential-Time Algorithms for Maximum Independent Set in P_t-Free and Broom-Free Graphs. Algorithmica, 81(2):421-438, 2019. URL: https://doi.org/10.1007/s00453-018-0479-5.
  3. Manuel Bodirsky, Jan Kára, and Barnaby Martin. The complexity of surjective homomorphism problems - a survey. Discrete Applied Mathematics, 160(12):1680-1690, 2012. URL: https://doi.org/10.1016/j.dam.2012.03.029.
  4. Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Oliver Schaudt, Maya Stein, and Mingxian Zhong. Three-Coloring and List Three-Coloring of Graphs Without Induced Paths on Seven Vertices. Combinatorica, 38(4):779-801, 2018. URL: https://doi.org/10.1007/s00493-017-3553-8.
  5. Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  6. Eglantine Camby and Oliver Schaudt. A New Characterization of P_k-Free Graphs. Algorithmica, 75(1):205-217, 2016. URL: https://doi.org/10.1007/s00453-015-9989-6.
  7. Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. Four-coloring P₆-free graphs. I. Extending an excellent precoloring. CoRR, abs/1802.02282, 2018. URL: http://arxiv.org/abs/1802.02282.
  8. Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. Four-coloring P₆-free graphs. II. Finding an excellent precoloring. CoRR, abs/1802.02283, 2018. URL: http://arxiv.org/abs/1802.02283.
  9. Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. Four-coloring P₆-free graphs. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1239-1256. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.76.
  10. Keith Edwards. The Complexity of Colouring Problems on Dense Graphs. Theor. Comput. Sci., 43:337-343, 1986. URL: https://doi.org/10.1016/0304-3975(86)90184-2.
  11. Jessica Enright, Lorna Stewart, and Gábor Tardos. On List Coloring and List Homomorphism of Permutation and Interval Graphs. SIAM J. Discrete Math., 28(4):1675-1685, 2014. URL: https://doi.org/10.1137/13090465X.
  12. Tomas Feder and Pavol Hell. List Homomorphisms to Reflexive Graphs. J. Comb. Theory, Ser. B, 72(2):236-250, 1998. URL: https://doi.org/10.1006/jctb.1997.1812.
  13. Tomás Feder, Pavol Hell, and Jing Huang. List Homomorphisms and Circular Arc Graphs. Combinatorica, 19(4):487-505, 1999. URL: https://doi.org/10.1007/s004939970003.
  14. Tomás Feder, Pavol Hell, Sulamita Klein, Loana Tito Nogueira, and Fábio Protti. List matrix partitions of chordal graphs. Theor. Comput. Sci., 349(1):52-66, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.030.
  15. Tomás Feder and Moshe Y. Vardi. Monotone monadic SNP and constraint satisfaction. In S. Rao Kosaraju, David S. Johnson, and Alok Aggarwal, editors, Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 612-622. ACM, 1993. URL: https://doi.org/10.1145/167088.167245.
  16. Anna Galluccio, Pavol Hell, and Jaroslav Nesetril. The complexity of H-colouring of bounded degree graphs. Discrete Mathematics, 222(1-3):101-109, 2000. URL: https://doi.org/10.1016/S0012-365X(00)00009-1.
  17. Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs. Journal of Graph Theory, 84(4):331-363, 2017. URL: https://doi.org/10.1002/jgt.22028.
  18. Petr A. Golovach, Daniël Paulusma, and Jian Song. Closing complexity gaps for coloring problems on H-free graphs. Inf. Comput., 237:204-214, 2014. URL: https://doi.org/10.1016/j.ic.2014.02.004.
  19. Carla Groenland, Karolina Okrasa, Paweł Rzążewski, Alex Scott, Paul Seymour, and Sophie Spirkl. H-colouring P_t-free graphs in subexponential time. Discrete Applied Mathematics, (to appear), 2019. Google Scholar
  20. Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, and Michal Pilipczuk. Polynomial-time algorithm for Maximum Weight Independent Set on P₆-free graphs. CoRR, abs/1707.05491, 2017. URL: http://arxiv.org/abs/1707.05491.
  21. Andrzej Grzesik, Tereza Klimošova, Marcin Pilipczuk, and Michal Pilipczuk. Polynomial-time algorithm for Maximum Weight Independent Set on P₆-free graphs. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1257-1271. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.77.
  22. Pavol Hell and Jaroslav Nesetril. Graphs and Homomorphisms. Oxford University Press, July 2004. URL: https://doi.org/10.1093/acprof:oso/9780198528173.001.0001.
  23. Pavol Hell and Jaroslav Nešetřil. On the complexity of H-coloring. J. Comb. Theory, Ser. B, 48(1):92-110, 1990. URL: https://doi.org/10.1016/0095-8956(90)90132-J.
  24. Chính T. Hoàng, Marcin Kamiński, Vadim V. Lozin, Joe Sawada, and Xiao Shu. Deciding k-Colorability of P₅-Free Graphs in Polynomial Time. Algorithmica, 57(1):74-81, 2010. URL: https://doi.org/10.1007/s00453-008-9197-8.
  25. Ian Holyer. The NP-Completeness of Edge-Coloring. SIAM J. Comput., 10(4):718-720, 1981. URL: https://doi.org/10.1137/0210055.
  26. Shenwei Huang. Improved complexity results on k-coloring P_t-free graphs. Eur. J. Comb., 51:336-346, 2016. URL: https://doi.org/10.1016/j.ejc.2015.06.005.
  27. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  28. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which Problems Have Strongly Exponential Complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  29. Jan Kratochvíl. Precoloring extension with fixed color bound. Acta Mathematica Universitatis Comenianae. New Series, 62, January 1993. Google Scholar
  30. Benoit Larose and Adrien Lemaître. List-homomorphism problems on graphs and arc consistency. Discrete Mathematics, 313(22):2525-2537, 2013. URL: https://doi.org/10.1016/j.disc.2013.07.018.
  31. Vadim V. Lozin and Marcin Kamiński. Coloring edges and vertices of graphs without short or long cycles. Contributions to Discrete Mathematics, 2(1), 2007. URL: http://cdm.ucalgary.ca/cdm/index.php/cdm/article/view/60.
  32. Gerhard J. Woeginger and Jirí Sgall. The complexity of coloring graphs without long induced paths. Acta Cybern., 15(1):107-117, 2001. URL: http://www.inf.u-szeged.hu/actacybernetica/edb/vol15n1/Woeginger_2001_ActaCybernetica.xml.
  33. Dmitriy Zhuk. A Proof of CSP Dichotomy Conjecture. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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