Skip to main content

The Chromatic Number of the Product of 14-Chromatic Graphs Can BE 13

Combinatorica volume 42, pages 301–308 (2022)Cite this article

Abstract

We show that for any n ≥ 13, there exist graphs with chromatic number larger than n whose product has chromatic number n. Our construction is an adaptation of the construction of counterexamples to Hedetniemi’s conjecture devised by Shitov, and adapted by Zhu to graphs with relatively small chromatic numbers. The new tools we introduce are graphs with minimal colourings that are “wide” in the sense of Simonyi and Tardos, and generalised Mycielskians to settle the case n = 13.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. S. Baum and M. Stiebitz: Coloring of graphs without short odd paths between vertices of the same color class (2005). Manuscript, 23 PP.

  2. S. A. Burr, P. Erdős and L. Lovász: On graphs of Ramsey type, Ars Combinatoria 1 (1976), 167–190.

    MathSciNet  MATH  Google Scholar 

  3. M. El-Zahar and N. Sauer: The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica 5 (1985), 121–126.

    MathSciNet  Article  Google Scholar 

  4. A. Gujgiczer and G. Simonyi: On multichromatic numbers of widely colorable graphs, manuscript (2021), arXiv:2102.03120.

  5. A. Gyárfás, T. Jensen and M. Stiebitz: On graphs with strongly independent colorclasses, J. Graph. Theory 46 (2004), 1–14.

    MathSciNet  Article  Google Scholar 

  6. G. Hahn and C. Tardif: Graph homomorphisms: structure and symmetry, Graph symmetry (Montreal, PQ, 1996), 107–166, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences 497, Kluwer Academic Publishers, Dordrecht, 1997.

    MATH  Google Scholar 

  7. H. Hajiabolhassan: On colorings of graph powers, Discrete Math. 309 (2009), 4299–4305.

    MathSciNet  Article  Google Scholar 

  8. X. He and Y. Wigderson: Hedetniemi’s conjecture is asympototically false, J. Combin. Theory Ser. B 146 (2021), 485–494.

    MathSciNet  Article  Google Scholar 

  9. S. Hedetniemi: Homomorphisms of graphs and automata, Technical Report 03105-44-T, University of Michigan, 1966.

  10. L. Lovász: Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory Ser. A 25 (1978), 319–324.

    MathSciNet  Article  Google Scholar 

  11. Y. Shitov: Counterexamples to Hedetniemi’s Conjecture, Ann. of Math. (2) 190 (2019), 663–667.

    MathSciNet  Article  Google Scholar 

  12. G. Simonyi and G. Tardos: Local chromatic number, Ky Fan’s Theorem and circular colorings, Combinatorica 26 (2006), 587–626.

    MathSciNet  Article  Google Scholar 

  13. C. Tardif and X. Zhu: A note on Hedetniemi’s conjecture, Stahl’s conjecture and the Poljak-Rödl function, Electron. J. Combin. 26 (2019), #P4.32.

    Article  Google Scholar 

  14. M. Wrochna: On inverse powers of graphs and topological implications of Hedetniemi’s conjecture, J. Combin. Theory Ser. B 139 (2019), 267–295.

    MathSciNet  Article  Google Scholar 

  15. M. Wrochna: Smaller counterexamples to Hedetniemi’s conjecture, manuscript (2020), arXiv:2012.13558.

  16. X. Zhu: A note on the Poljak—Rödl function, Electron. J. Combin. 27 (2020), #P3.2.

    Article  Google Scholar 

  17. X. Zhu: Relatively small counterexamples to Hedetniemi’s conjecture, J. Combin. Theory Ser. B 146 (2021), 141–150.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada

    Claude Tardif

Authors
  1. Claude Tardif
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claude Tardif.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tardif, C. The Chromatic Number of the Product of 14-Chromatic Graphs Can BE 13. Combinatorica 42, 301–308 (2022). https://doi.org/10.1007/s00493-021-4781-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-021-4781-5

Mathematics Subject Classification (2010)

  • 05C15