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Counterexamples to Hedetniemi’s Conjecture with Large Fractional Chromatic Numbers

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Abstract

We show that the inequality

$$\begin{aligned} \chi (G\times H) < \min \{ \chi _f(G), \chi (H)\} \end{aligned}$$

can happen when \(\chi (G\times H) = 43\), improving on the lowest previously known value \(\chi (G\times H) = 125\).

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. The values 2, 22, 25, 34, 49, 58, 61, 81 were used originally, but here these are multiplied by 38 modulo 83 to simplify the presentation.

References

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Gao, G., Zhu, X.: Star-extremal graphs and lexicographic product. Discrete Math. 152, 147–156 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. The GAP Group.: GAP–Groups, algorithms, and programming, Version 4.11.1. (2021). https://www.gap-system.org

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

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Soicher, L.H.: GRAPE, graph algorithms using permutation groups-a refereed GAP package, Version 4.8.5. (2021). (https://gap-packages.github.io/grape)

  9. Tardif, C.: The chromatic number of the product of 14-chromatic graphs can be 13. Combinatorica 42, 301–308 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. Turzík, D.: A note on chromatic number of direct product of graphs. Comment. Math. Univ. Carolin. 24, 461–463 (1983)

    MathSciNet  MATH  Google Scholar 

  11. Wrochna, M.: Smaller counterexamples to Hedetniemi’s conjecture. (2022). https://arxiv.org/abs/2012.13558 [math.CO]

  12. Zhu, X.: The fractional version of Hedetniemi’s conjecture is true. Eur. J. Combin. 32, 1168–1175 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhu, X.: A note on the Poljak-Rödl function. Electron. J. Combin. 27(3.2), 4 (2020)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Funding

The author declares that no funds, grants, or other support were received during the preparation of this manuscript. The author has no relevant financial or non-financial interests to disclose.

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

    Claude Tardif

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  1. Claude Tardif
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Correspondence to Claude Tardif.

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Tardif, C. Counterexamples to Hedetniemi’s Conjecture with Large Fractional Chromatic Numbers. Graphs and Combinatorics 38, 171 (2022). https://doi.org/10.1007/s00373-022-02576-x

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  • DOI: https://doi.org/10.1007/s00373-022-02576-x

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