Skip to main content
SpringerLink
Log in

Approximately Coloring Graphs Without Long Induced Paths

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on t vertices, for fixed t. We propose an algorithm that, given a 3-colorable graph without an induced path on t vertices, computes a coloring with \(\max \left\{ 5,2\left\lceil \frac{t-1}{2}\right\rceil -2\right\} \) many colors. If the input graph is triangle-free, we only need \(\max \left\{ 4,\left\lceil \frac{t-1}{2}\right\rceil +1\right\} \) many colors. The running time of our algorithm is \(O((3^{t-2}+t^2)m+n)\) if the input graph has n vertices and m edges.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brakensiek, J., Guruswami, V.: New hardness results for graph and hypergraph colorings. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 50. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2016)

  2. Bonomo, F., Chudnovsky, M., Maceli, P., Schaudt, O., Stein, M., Zhong, M.: Three-coloring and list three-coloring graphs without induced paths on seven vertices. Combinatorica 38(4), 779–801 (2018)

  3. Chlamtac, E.: Approximation algorithms using hierarchies of semidefinite programming relaxations. In: 48th Annual IEEE Symposium on Foundations of Computer Science (2007)

  4. Chudnovsky, M., Spirkl, S., Zhong, M.: Four-coloring \(P_6\)-free graphs. I. Extending an excellent precoloring. arXiv preprint arXiv:1802.02282 (2018)

  5. Chudnovsky, M., Spirkl, S., Zhong, M.: Four-coloring \(P_6\)-free graphs. II. Finding an excellent precoloring. arXiv preprint arXiv:1802.02283 (2018)

  6. Chuzhoy, J.: Private communication (2015)

  7. Dinur, I., Mossel, E., Regev, O.: Conditional hardness for approximate coloring. SIAM J. Comput. 39, 843–873 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Edwards, K.: The complexity of colouring problems on dense graphs. Theor. Comput. Sci. 43, 337–343 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs. Congressus Numerantium 26, 125–157 (1979)

    MATH  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: A Guide to the Theory of NP-Completeness. WH Freemann, New York (1979)

    MATH  Google Scholar 

  11. Groenland, C., Okrasa, K., Rzążewski, P., Scott, A., Seymour, P., Spirkl, S.: \( H \)-colouring \( P_t \)-free graphs in subexponential time. arXiv preprint arXiv:1803.05396 (2018)

  12. Golovach, P.A., Johnson, M., Paulusma, D., Song, J.: A survey on the computational complexity of colouring graphs with forbidden subgraphs. J. Graph Theory 84(4), 331–363 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gyárfás, A.: Problems from the world surrounding perfect graphs. Appl. Math. 19(3–4), 413–441 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Hoàng, C.T., Kamiński, M., Lozin, V., Sawada, J., Shu, X.: Deciding \(k\)-colorability of \(P_5\)-free graphs in polynomial time. Algorithmica 57(1), 74–81 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10, 718–720 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, S.: Improved complexity results on \(k\)-coloring \(P_t\)-free graphs. Eur. J. Comb. 51, 336–346 (2016)

    Article  MATH  Google Scholar 

  17. Huang, S., Johnson, M., Paulusma, D.: Narrowing the complexity gap for colouring \((C_s, P_t)\)-free graphs. Comput. J. 58(11), 3074–3088 (2015)

    Article  MATH  Google Scholar 

  18. Kamiński, M., Lozin, V.: Coloring edges and vertices of graphs without short or long cycles. Contrib. Discrete Math. 2, 61–66 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  20. Kawarabayashi, K., Thorup, M.: Coloring 3-Colorable Graphs with \(o (n^{1/5})\) Colors. LIPIcs-Leibniz International Proceedings in Informatics, vol. 25. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Wadern (2014)

    MATH  Google Scholar 

  21. Král, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. Proc. WG 2001, 254–262 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Leven, D., Galil, Z.: NP-completeness of finding the chromatic index of regular graphs. J. Algorithms 4, 35–44 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Vizing, V.G.: Coloring the vertices of a graph in prescribed colors. Diskret. Analiz 29(3), 3–10 (1976)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

We are thankful to Paul Seymour for many helpful discussions. We thank Stefan Hougardy for pointing out [20] to us. This material is based upon work supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under Grant No. W911NF-16-1-0404.

Author information

Author notes

  1. Oliver Schaudt

    Present address: RWTH Aachen, Aachen, Germany

  2. Sophie Spirkl

    Present address: Rutgers University, Piscataway, NJ, 08854, USA

  3. Mingxian Zhong

    Present address: Lehman College, CUNY, Bronx, NY, 10468, USA

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Maria Chudnovsky & Sophie Spirkl

  2. Universität zu Köln, Cologne, Germany

    Oliver Schaudt

  3. Universidad de Chile, Santiago, Chile

    Maya Stein

  4. Columbia University, New York, NY, 10027, USA

    Mingxian Zhong

Authors
  1. Maria Chudnovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oliver Schaudt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sophie Spirkl
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Maya Stein
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mingxian Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sophie Spirkl.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

An extended abstract of the paper has previously appeared in Bodlaender H., Woeginger G. (eds): Graph-Theoretic Concepts in Computer Science (WG) 2017. Lecture Notes in Computer Science, vol 10520. Springer, Cham.

The first author was supported by National Science Foundation grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. The fourth author was supported by Fondecyt Grants 1140766 and 1180830, by CMM-Basal AFB 170001, and by Millennium Nucleus Information and Coordination in Networks.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chudnovsky, M., Schaudt, O., Spirkl, S. et al. Approximately Coloring Graphs Without Long Induced Paths. Algorithmica 81, 3186–3199 (2019). https://doi.org/10.1007/s00453-019-00577-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-019-00577-6

Keywords

Mathematics Subject Classification

Search

Navigation