Complexity of Computing the Anti-Ramsey Numbers for Paths

Authors Saeed Akhoondian Amiri , Alexandru Popa , Mohammad Roghani , Golnoosh Shahkarami , Reza Soltani , Hossein Vahidi



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2020.6.pdf
  • Filesize: 0.95 MB
  • 14 pages

Document Identifiers

Author Details

Saeed Akhoondian Amiri
  • University of Cologne, Germany
Alexandru Popa
  • University of Bucharest, Romania
  • National Institute of Research and Development in Informatics, Bucharest, Romania
Mohammad Roghani
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
  • Sharif University of Technology, Teheran, Iran
Golnoosh Shahkarami
  • MPI for Informatics, Saarland Informatics Campus, Graduate School of Computer Science, Saarbrücken, Germany
Reza Soltani
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
  • Sharif University of Technology, Teheran, Iran
Hossein Vahidi
  • MPI for Informatics, Saarland Informatics Campus, Graduate School of Computer Science, Saarbrücken, Germany

Cite AsGet BibTex

Saeed Akhoondian Amiri, Alexandru Popa, Mohammad Roghani, Golnoosh Shahkarami, Reza Soltani, and Hossein Vahidi. Complexity of Computing the Anti-Ramsey Numbers for Paths. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.6

Abstract

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar(G,H) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar(G,P_k), where P_k is a path of length k. First, we observe that when k is close to n, the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar(G,P_k) for every integer k > 2 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating ar(G,P₃) to a factor of n^{-1/2 - ε} is hard already in 3-partite graphs, unless P = NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. Along this line, we first introduce the notion of color connected coloring, and, employing this structural property, we obtain a linear time algorithm to compute ar(G,P_k), for every integer k, when the host graph, G, is a tree.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Graph algorithms analysis
Keywords
  • Coloring
  • Anti-Ramsey
  • Approximation
  • NP-hard
  • Algorithm
  • ETH

Metrics

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

References

  1. A. Adamaszek and A. Popa. Approximation and hardness results for the maximum edge q-coloring problem. J. Discrete Algorithms, 38-41:1-8, 2016. Google Scholar
  2. Anna Adamaszek and Alexandru Popa. Approximation and hardness results for the maximum edge q-coloring problem. In ISAAC (2), pages 132-143, 2010. Google Scholar
  3. Saeed Akhoondian Amiri, Alexandru Popa, Mohammad Roghani, Golnoosh Shahkarami, Reza Soltani, and Hossein Vahidi. Complexity of computing the anti-ramsey numbers for paths, 2018. URL: http://arxiv.org/abs/1810.08004.
  4. Maria Axenovich and Tao Jiang. Anti-Ramsey numbers for small complete bipartite graphs. Ars Comb., 73, 2004. Google Scholar
  5. Maria Axenovich, Tao Jiang, and André Kündgen. Bipartite anti-Ramsey numbers of cycles. Journal of Graph Theory, 47(1):9-28, 2004. Google Scholar
  6. Aart Blokhuis, Ralph J. Faudree, András Gyárfás, and Miklós Ruszinkó. Anti-Ramsey colorings in several rounds. J. Comb. Theory, Ser. B, 82(1):1-18, 2001. URL: https://doi.org/10.1006/jctb.2000.2016.
  7. Csilla Bujtás, E. Sampathkumar, Zsolt Tuza, Charles Dominic, and L. Pushpalatha. 3-consecutive edge coloring of a graph. Discrete Mathematics, 312(3):561-573, 2012. URL: https://doi.org/10.1016/j.disc.2011.04.006.
  8. He Chen, Xueliang Li, and Jianhua Tu. Complete solution for the rainbow numbers of matchings. Discrete Mathematics, 309(10):3370-3380, 2009. URL: https://doi.org/10.1016/j.disc.2008.10.002.
  9. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  10. P. Erdös, M. Simonovits, and V. T. Sós. Anti-Ramsey theorems. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdös on his 60th birthday), pages 633-643, 1975. Google Scholar
  11. W. Feng, L. Zhang, W. Qu, and H. Wang. Approximation algorithms for maximum edge coloring problem. In TAMC, pages 646-658, 2007. Google Scholar
  12. Wangsen Feng, Ping Chen, and Bei Zhang. Approximate maximum edge coloring within factor 2: a further analysis. In ISORA, pages 182-189, 2008. Google Scholar
  13. Wangsen Feng, Li'ang Zhang, and Hanpin Wang. Approximation algorithm for maximum edge coloring. Theoretical Computer Science, 410(11):1022-1029, 2009. Google Scholar
  14. Alan Frieze and Bruce Reed. Polychromatic hamilton cycles. Discrete Mathematics, 118(1):69-74, 1993. Google Scholar
  15. Shinya Fujita, Colton Magnant, and Kenta Ozeki. Rainbow generalizations of Ramsey theory: A dynamic survey. Theory and Applications of Graphs, 0, 2014. Google Scholar
  16. Wayne Goddard and Honghai Xu. Vertex colorings without rainbow subgraphs. Discussiones Mathematicae Graph Theory, 36(4):989-1005, 2016. URL: https://doi.org/10.7151/dmgt.1896.
  17. Ruth Haas and Michael Young. The anti-Ramsey number of perfect matching. Discrete Mathematics, 312(5):933-937, 2012. URL: https://doi.org/10.1016/j.disc.2011.10.017.
  18. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  19. Tao Jiang. Edge-colorings with no large polychromatic stars. Graphs and Combinatorics, 18(2):303-308, May 2002. Google Scholar
  20. Tao Jiang and Douglas B. West. Edge-colorings of complete graphs that avoid polychromatic trees. Discrete Mathematics, 274(1-3):137-145, 2004. URL: https://doi.org/10.1016/j.disc.2003.09.002.
  21. J.J. Montellano-Ballesteros and V. Neumann-Lara. An anti-Ramsey theorem on cycles. Graphs and Combinatorics, 21(3):343-354, September 2005. Google Scholar
  22. Ingo Schiermeyer. Rainbow numbers for matchings and complete graphs. Discrete Mathematics, 286(1-2):157-162, 2004. Google Scholar
  23. Ingo Schiermeyer. Rainbow colourings. Invited papers from RIMS, Kyoto University, 2007. Google Scholar
  24. Miklós Simonovits and Vera T. Sós. On restricted colourings of K_n. Combinatorica, 4(1):101-110, March 1984. Google Scholar
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