Document Open Access Logo

C_{2k+1}-Coloring of Bounded-Diameter Graphs

Author Marta Piecyk

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2024.78.pdf
  • Filesize: 0.86 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph coloring
Keywords
  • graph homomorphism
  • odd cycles
  • diameter

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

For a fixed graph H, in the graph homomorphism problem, denoted by Hom(H), we are given a graph G and we have to determine whether there exists an edge-preserving mapping φ: V(G) → V(H). Note that Hom(C₃), where C₃ is the cycle of length 3, is equivalent to 3-Coloring. The question of whether 3-Coloring is polynomial-time solvable on diameter-2 graphs is a well-known open problem. In this paper we study the Hom(C_{2k+1}) problem on bounded-diameter graphs for k ≥ 2, so we consider all other odd cycles than C₃. We prove that for k ≥ 2, the Hom(C_{2k+1}) problem is polynomial-time solvable on diameter-(k+1) graphs - note that such a result for k = 1 would be precisely a polynomial-time algorithm for 3-Coloring of diameter-2 graphs. Furthermore, we give subexponential-time algorithms for diameter-(k+2) and -(k+3) graphs.
We complement these results with a lower bound for diameter-(2k+2) graphs - in this class of graphs the Hom(C_{2k+1}) problem is NP-hard and cannot be solved in subexponential-time, unless the ETH fails.
Finally, we consider another direction of generalizing 3-Coloring on diameter-2 graphs. We consider other target graphs H than odd cycles but we restrict ourselves to diameter 2. We show that if H is triangle-free, then Hom(H) is polynomial-time solvable on diameter-2 graphs.

Cite As Get BibTex

Marta Piecyk. C_{2k+1}-Coloring of Bounded-Diameter Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.78

Author Details

Marta Piecyk
  • Warsaw University of Technology, Poland

Funding

  • Piecyk, Marta: This research was funded by Polish National Science Centre, grant no. 2022/45/N/ST6/00237.

Acknowledgements

We are grateful to Paweł Rzążewski for inspiring and fruitful discussions.

References

  1. Laurent Beaudou, Florent Foucaud, and Reza Naserasr. Smallest c_2l+1-critical graphs of odd-girth 2k+1. Discret. Appl. Math., 319:564-575, 2022. URL: https://doi.org/10.1016/J.DAM.2021.08.040.
  2. Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Independent feedback vertex sets for graphs of bounded diameter. Inf. Process. Lett., 131:26-32, 2018. URL: https://doi.org/10.1016/J.IPL.2017.11.004.
  3. Christoph Brause, Petr A. Golovach, Barnaby Martin, Pascal Ochem, Daniël Paulusma, and Siani Smith. Acyclic, star, and injective colouring: Bounding the diameter. Electron. J. Comb., 29(2), 2022. URL: https://doi.org/10.37236/10738.
  4. Andrei A. Bulatov and Amirhossein Kazeminia. Complexity classification of counting graph homomorphisms modulo a prime number. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1024-1037. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520075.
  5. Jin-Yi Cai and Ashwin Maran. The complexity of counting planar graph homomorphisms of domain size 3. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 1285-1297. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585173.
  6. Victor A. Campos, Guilherme de C. M. Gomes, Allen Ibiapina, Raul Lopes, Ignasi Sau, and Ana Silva. Coloring problems on bipartite graphs of small diameter. Electron. J. Comb., 28(2):2, 2021. URL: https://doi.org/10.37236/9931.
  7. Rajesh Chitnis, László Egri, and Dániel Marx. List h-coloring a graph by removing few vertices. Algorithmica, 78(1):110-146, 2017. URL: https://doi.org/10.1007/S00453-016-0139-6.
  8. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 210-223. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055502.
  9. Michal Dębski, Marta Piecyk, and Paweł Rzążewski. Faster 3-coloring of small-diameter graphs. SIAM J. Discret. Math., 36(3):2205-2224, 2022. URL: https://doi.org/10.1137/21M1447714.
  10. Oliver Ebsen and Mathias Schacht. Homomorphism thresholds for odd cycles. Comb., 40(1):39-62, 2020. URL: https://doi.org/10.1007/S00493-019-3920-8.
  11. 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.
  12. Tomás Feder, Pavol Hell, and Jing Huang. List homomorphisms and circular arc graphs. Comb., 19(4):487-505, 1999. URL: https://doi.org/10.1007/S004939970003.
  13. Tomás Feder, Pavol Hell, and Jing Huang. Bi-arc graphs and the complexity of list homomorphisms. J. Graph Theory, 42(1):61-80, 2003. URL: https://doi.org/10.1002/JGT.10073.
  14. Jacob Focke, Dániel Marx, and Paweł Rzążewski. Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 431-458. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.22.
  15. Robert Ganian, Thekla Hamm, Viktoriia Korchemna, Karolina Okrasa, and Kirill Simonov. The fine-grained complexity of graph homomorphism parameterized by clique-width. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 66:1-66:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ICALP.2022.66.
  16. A. M. H. Gerards. Homomorphisms of graphs into odd cycles. J. Graph Theory, 12(1):73-83, 1988. URL: https://doi.org/10.1002/JGT.3190120108.
  17. Carla Groenland, Isja Mannens, Jesper Nederlof, Marta Piecyk, and Paweł Rzążewski. Towards tight bounds for the graph homomorphism problem parameterized by cutwidth via asymptotic rank parameters. CoRR, abs/2312.03859, 2023. URL: https://doi.org/10.48550/arXiv.2312.03859.
  18. Pavol Hell and Jaroslav Nešetřil. On the complexity of h-coloring. Journal of Combinatorial Theory, Series B, 48(1):92-110, 1990. URL: https://doi.org/10.1016/0095-8956(90)90132-J.
  19. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/JCSS.2000.1727.
  20. Tereza Klimosová and Vibha Sahlot. 3-coloring c_4 or c_3-free diameter two graphs. In Pat Morin and Subhash Suri, editors, Algorithms and Data Structures - 18th International Symposium, WADS 2023, Montreal, QC, Canada, July 31 - August 2, 2023, Proceedings, volume 14079 of Lecture Notes in Computer Science, pages 547-560. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-38906-1_36.
  21. Aleksej Dmitrievich Korshunov. On the diameter of graphs. Soviet Math, 12:302:305, 1971. Google Scholar
  22. Hong-Jian Lai. Unique graph homomorphisms onto odd cycles, II. J. Comb. Theory, Ser. B, 46(3):363-376, 1989. URL: https://doi.org/10.1016/0095-8956(89)90056-7.
  23. Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring h-free graphs of bounded diameter. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 14:1-14:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.MFCS.2019.14.
  24. Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring graphs of bounded diameter in the absence of small cycles. Discret. Appl. Math., 314:150-161, 2022. URL: https://doi.org/10.1016/J.DAM.2022.02.026.
  25. George B. Mertzios and Paul G. Spirakis. Algorithms and almost tight results for 3-colorability of small diameter graphs. Algorithmica, 74(1):385-414, 2016. URL: https://doi.org/10.1007/S00453-014-9949-6.
  26. Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Full complexity classification of the list homomorphism problem for bounded-treewidth graphs. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 74:1-74:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ESA.2020.74.
  27. Karolina Okrasa and Paweł Rzążewski. Fine-grained complexity of the graph homomorphism problem for bounded-treewidth graphs. SIAM J. Comput., 50(2):487-508, 2021. URL: https://doi.org/10.1137/20M1320146.
  28. Christos H. Papadimitriou. Computational complexity. Addison-Wesley, 1994. Google Scholar
  29. Marta Piecyk. C_2k+1-coloring of bounded-diameter graphs. CoRR, abs/2403.06694, 2024. URL: https://doi.org/10.48550/arXiv.2403.06694.
  30. Sebastian Schnettler. A structured overview of 50 years of small-world research. Soc. Networks, 31(3):165-178, 2009. URL: https://doi.org/10.1016/J.SOCNET.2008.12.004.
  31. Petra Sparl and Janez Zerovnik. Homomorphisms of hexagonal graphs to odd cycles. Discret. Math., 283(1-3):273-277, 2004. URL: https://doi.org/10.1016/J.DISC.2003.12.012.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact