Skip to main content
SpringerLink
Log in

Maximum Edge Colouring Problem On Graphs That Exclude a Fixed Minor

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

Included in the following conference series:

  • 326 Accesses

Abstract

The maximum edge colouring problem considers the maximum colour assignment to edges of a graph under the condition that every vertex has at most a fixed number of distinct coloured edges incident on it. If that fixed number is q we call the colouring a maximum edge q-colouring. The problem models a non-overlapping frequency channel assignment question on wireless networks. The problem has also been studied from a purely combinatorial perspective in the graph theory literature.

We study the question when the input graph is sparse. We show the problem remains \(\texttt {NP}\)-hard on 1-apex graphs. We also show that there exists \(\texttt {PTAS}\) for the problem on minor-free graphs. The \(\texttt {PTAS}\) is based on a recently developed Baker game technique for proper minor-closed classes, thus avoiding the need to use any involved structural results. This further pushes the Baker game technique beyond the problems expressible in the first-order logic.

Supported by project 22-17398S (Flows and cycles in graphs on surfaces) of Czech Science Foundation

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adamaszek, A., Popa, A.: Approximation and hardness results for the maximum edge q-coloring problem. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 132–143. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17514-5_12

    Chapter  MATH  Google Scholar 

  2. Adamaszek, A., Popa, A.: Approximation and hardness results for the maximum edge Q-coloring problem. J. Discret. Algorithms 38-41, 1–8 (2016). https://doi.org/10.1016/j.jda.2016.09.003

  3. Alon, N., Seymour, P.D., Thomas, R.: A separator theorem for graphs with an excluded minor and its applications. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, 13–17 May 1990, Baltimore, Maryland, USA, pp. 293–299. ACM (1990). https://doi.org/10.1145/100216.100254

  4. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994). https://doi.org/10.1145/174644.174650

    Article  MathSciNet  MATH  Google Scholar 

  5. Cabello, S., Gajser, D.: Simple PTAS’s for families of graphs excluding a minor. Discret. Appl. Math. 189, 41–48 (2015). https://doi.org/10.1016/j.dam.2015.03.004

    Article  MathSciNet  MATH  Google Scholar 

  6. Chandran, L.S., Hashim, T., Jacob, D., Mathew, R., Rajendraprasad, D., Singh, N.: New bounds on the anti-Ramsey numbers of star graphs. CoRR abs/1810.00624 (2018). https://arxiv.org/abs/1810.00624

  7. Chandran, L.S., Lahiri, A., Singh, N.: Improved approximation for maximum edge colouring problem. Discrete Appl. Math. 319, 42–52 (2022). https://doi.org/10.1016/j.dam.2021.05.017

    Article  MathSciNet  MATH  Google Scholar 

  8. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990). https://doi.org/10.1016/0890-5401(90)90043-H

    Article  MathSciNet  MATH  Google Scholar 

  9. Dawar, A., Grohe, M., Kreutzer, S., Schweikardt, N.: Approximation schemes for first-order definable optimisation problems. In: 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12–15 August 2006, Seattle, WA, USA, Proceedings, pp. 411–420. IEEE Computer Society (2006). https://doi.org/10.1109/LICS.2006.13

  10. Demaine, E.D., Hajiaghayi, M.T.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, 23–25 January 2005, pp. 590–601. SIAM (2005). https://dl.acm.org/citation.cfm?id=1070432.1070514

  11. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  12. Dvorák, Z.: Thin graph classes and polynomial-time approximation schemes. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, 7–10 January 2018, pp. 1685–1701. SIAM (2018). https://doi.org/10.1137/1.9781611975031.110

  13. Dvorák, Z.: Baker game and polynomial-time approximation schemes. In: Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, 5–8 January 2020, pp. 2227–2240. SIAM (2020). https://doi.org/10.1137/1.9781611975994.137

  14. Erdös, P., Simonovits, M., Sós, V.T.: Anti-Ramsey theorems. Infinite Finite Sets (Colloquium, Keszthely, 1973; dedicated to P. Erdös on his 60th birthday) 10(II), 633–643 (1975)

    Google Scholar 

  15. Feng, W., Zhang, L., Wang, H.: Approximation algorithm for maximum edge coloring. Theor. Comput. Sci. 410(11), 1022–1029 (2009). https://doi.org/10.1016/j.tcs.2008.10.035

    Article  MathSciNet  MATH  Google Scholar 

  16. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Bidimensionality and EPTAS. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, 23–25 January 2011, pp. 748–759. SIAM (2011). https://doi.org/10.1137/1.9781611973082.59

  17. Fujita, S., Magnant, C., Ozeki, K.: Rainbow generalizations of Ramsey theory: a survey. Graphs Combin. 26(1), 1–30 (2010). https://doi.org/10.1007/s00373-010-0891-3

    Article  MathSciNet  MATH  Google Scholar 

  18. Gorgol, I., Lazuka, E.: Rainbow numbers for small stars with one edge added. Discuss. Math. Graph Theory 30(4), 555–562 (2010). https://doi.org/10.7151/dmgt.1513

    Article  MathSciNet  MATH  Google Scholar 

  19. Goyal, P., Kamat, V., Misra, N.: On the parameterized complexity of the maximum edge 2-coloring problem. In: Mathematical Foundations of Computer Science 2013–38th International Symposium, MFCS 2013, Klosterneuburg, Austria, 26–30 August 2013, pp. 492–503 (2013). https://doi.org/10.1007/978-3-642-40313-2_44

  20. Har-Peled, S., Quanrud, K.: Approximation algorithms for polynomial-expansion and low-density graphs. SIAM J. Comput. 46(6), 1712–1744 (2017). https://doi.org/10.1137/16M1079336

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiang, T.: Edge-colorings with no large polychromatic stars. Graphs Combin. 18(2), 303–308 (2002). https://doi.org/10.1007/s003730200022

    Article  MathSciNet  MATH  Google Scholar 

  22. Klein, P.N., Plotkin, S.A., Rao, S.: Excluded minors, network decomposition, and multicommodity flow. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, 16–18 May 1993, San Diego, CA, USA, pp. 682–690. ACM (1993). https://doi.org/10.1145/167088.167261

  23. Kodialam, M.S., Nandagopal, T.: Characterizing the capacity region in multi-radio multi-channel wireless mesh networks. In: Proceedings of the 11th Annual International Conference on Mobile Computing and Networking, MOBICOM 2005, Cologne, Germany, 28 August–2 September 2005, pp. 73–87. ACM (2005), https://doi.org/10.1145/1080829.1080837

  24. Manoussakis, Y., Spyratos, M., Tuza, Z., Voigt, M.: Minimal colorings for properly colored subgraphs. Graphs Combin. 12(1), 345–360 (1996). https://doi.org/10.1007/BF01858468

    Article  MathSciNet  MATH  Google Scholar 

  25. Montellano-Ballesteros, J.J.: On totally multicolored stars. J. Graph Theory 51(3), 225–243 (2006). https://doi.org/10.1002/jgt.20140

    Article  MathSciNet  MATH  Google Scholar 

  26. Montellano-Ballesteros, J.J., Neumann-Lara, V.: An anti-Ramsey theorem. Combinatorica 22(3), 445–449 (2002). https://doi.org/10.1007/s004930200023

    Article  MathSciNet  MATH  Google Scholar 

  27. Raniwala, A., Chiueh, T.: Architecture and algorithms for an IEEE 802.11-based multi-channel wireless mesh network. In: INFOCOM 2005. 24th Annual Joint Conference of the IEEE Computer and Communications Societies, 13–17 March 2005, Miami, FL, USA, pp. 2223–2234 (2005). https://doi.org/10.1109/INFCOM.2005.1498497

  28. Robertson, N., Seymour, P.D.: Graph minors. XVI. Excluding a non-planar graph. J. Combin. Theory Ser. B 89(1), 43–76 (2003). https://doi.org/10.1016/S0095-8956(03)00042-X

    Article  MathSciNet  MATH  Google Scholar 

  29. Schiermeyer, I.: Rainbow numbers for matchings and complete graphs. Discret. Math. 286(1–2), 157–162 (2004). https://doi.org/10.1016/j.disc.2003.11.057

    Article  MathSciNet  MATH  Google Scholar 

  30. Sen, A., Murthy, S., Ganguly, S., Bhatnagar, S.: An interference-aware channel assignment scheme for wireless mesh networks. In: Proceedings of IEEE International Conference on Communications, ICC 2007, Glasgow, Scotland, UK, 24–28 June 2007, pp. 3471–3476. IEEE (2007). https://doi.org/10.1109/ICC.2007.574

  31. Simonovits, M., Sós, V.: On restricted colourings of \(K_n\). Combinatorica 4(1), 101–110 (1984). https://doi.org/10.1007/BF02579162

    Article  MathSciNet  MATH  Google Scholar 

  32. Tovey, C.A.: A simplified NP-complete satisfiability problem. Discret. Appl. Math. 8(1), 85–89 (1984). https://doi.org/10.1016/0166-218X(84)90081-7

    Article  MathSciNet  MATH  Google Scholar 

  33. Wan, P., Al-dhelaan, F., Jia, X., Wang, B., Xing, G.: Maximizing network capacity of MPR-capable wireless networks. In: 2015 IEEE Conference on Computer Communications, INFOCOM 2015, Kowloon, Hong Kong, 26 April–1 May 2015, pp. 1805–1813. IEEE (2015). https://doi.org/10.1109/INFOCOM.2015.7218562

  34. Wan, P., Cheng, Y., Wang, Z., Yao, F.F.: Multiflows in multi-channel multi-radio multihop wireless networks. In: INFOCOM 2011. 30th IEEE International Conference on Computer Communications, 10–15 April 2011, Shanghai, China, pp. 846–854. IEEE (2011). https://doi.org/10.1109/INFCOM.2011.5935308

Download references

Acknowledgement

The second author likes to thank Benjamin Moore, Jatin Batra, Sandip Banerjee and Siddharth Gupta for helpful discussions on this project. He also likes to thank the organisers of Homonolo for providing a nice and stimulating research environment.

Author information

Authors and Affiliations

  1. Charles University, 11800, Prague, Czech Republic

    Zdeněk Dvořák

  2. Heinrich Heine University, 40225, Düsseldorf, Germany

    Abhiruk Lahiri

Authors
  1. Zdeněk Dvořák
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abhiruk Lahiri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abhiruk Lahiri .

Editor information

Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dvořák, Z., Lahiri, A. (2023). Maximum Edge Colouring Problem On Graphs That Exclude a Fixed Minor. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-43380-1_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43379-5

  • Online ISBN: 978-3-031-43380-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation