1-Subdivisions, the Fractional Chromatic Number and the Hall Ratio

Abstract

The Hall ratio of a graph G is the maximum of |V(H)|/α(H) over all subgraphs H of G. It is easy to see that the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by subdividing each edge exactly once).

  • For every c > 0, every graph of sufficiently large average degree contains as a subgraph the 1-subdivision of a graph of fractional chromatic number at least c.

  • For every d > 0, there exists a graph G of average degree at least d such that every graph whose 1-subdivision appears as a subgraph of G has Hall ratio at most 18.

We also discuss the consequences of these results in the context of graph classes with bounded expansion.

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References

  1. [1]

    J. Barnett: The fractional chromatic number and the Hall ratio, PhD thesis, Auburn University, 2016.

  2. [2]

    A. Blumenthal, B. Lidický, R. R. Martin, S. Norin, F. Pfender and J. Volec: Counterexamples to a conjecture of Harris on Hall ratio, arXiv:1811.11116 (2018).

  3. [3]

    M. Cropper, A. Gyárfás and J. Lehel: Hall ratio of the Mycielski graphs, Discrete mathematics 306 (2006), 1988–1990.

    MathSciNet  Article  Google Scholar 

  4. [4]

    A. Daneshgar, A. Hilton and P. Johnson: Relations among the fractional chromatic, choice, Hall, and Hall-condition numbers of simple graphs, Discrete Mathematics 241 (2001), 189–199.

    MathSciNet  Article  Google Scholar 

  5. [5]

    Z. Dvořák: Asymptotical structure of combinatorial objects, PhD thesis, Charles University in Prague, 2007.

  6. [6]

    Z. Dvořák: On forbidden subdivision characterizations of graph classes, European Journal of Combinatorics 29 (2008), 1321–1332.

    MathSciNet  Article  Google Scholar 

  7. [7]

    D. G. Harris: Some results on chromatic number as a function of triangle count, arXiv:1604.00438 (2016).

  8. [8]

    P. D. Johnson Jr: The fractional chromatic number, the Hall ratio, and the lexicographic product, Discrete Mathematics 309 (2009), 4746–4749.

    MathSciNet  Article  Google Scholar 

  9. [9]

    L. Lovász: Kneser’s conjecture, chromatic number, and homotopy, Journal of Combinatorial Theory, Series A 25 (1978), 319–324.

    MathSciNet  Article  Google Scholar 

  10. [10]

    J. Nešetřil: A combinatorial classic-sparse graphs with high chromatic number, in: Erdős Centennial, Springer, 383–407, 2013.

  11. [11]

    J. Nešetřil and P. Ossona de Mendez: Grad and classes with bounded expansion I. Decompositions., European J. Combin. 29 (2008), 760–776.

    MathSciNet  Article  Google Scholar 

  12. [12]

    J. Nešetřil and P. Ossona de Mendez: On nowhere dense graphs, European J. Combin. 32 (2011), 600–617.

    MathSciNet  Article  Google Scholar 

  13. [13]

    J. Nešetřil and P. Ossona de Mendez: Sparsity (Graphs, Structures, and Algorithms), vol. 28 of Algorithms and Combinatorics, Springer, 2012.

  14. [14]

    S. Plotkin, S. Rao and W. D. Smith: Shallow excluded minors and improved graph decompositions, in: Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms, Society for Industrial and Applied Mathematics, 1994, 462–470.

  15. [15]

    L. Pyber, V. Rödl and E. Szemerédi: Dense graphs without 3-regular subgraphs, J. Combin. Theory, Ser. B 63 (1995), 41–54.

    MathSciNet  Article  Google Scholar 

  16. [16]

    E. R. Scheinerman and D. H. Ullman: Fractional graph theory, Dover Publications Inc., Mineola, NY, 2011.

    Google Scholar 

Download references

Acknowledgments

The results on Hall ratio were inspired by discussions taking place during the first Southwestern German Workshop on Graph Theory in Karlsruhe. We would like to thank the reviewers, whose suggestions helped us improve the presentation of the paper.

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Correspondence to Zdenĕk Dvořák.

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Supported by the project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foundation.

Supported by grant ERCCZ LL-1201 and by the European Associated Laboratory “Structures in Combinatorics” (LEA STRUCO).

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Dvořák, Z., de Mendez, P.O. & Wu, H. 1-Subdivisions, the Fractional Chromatic Number and the Hall Ratio. Combinatorica 40, 759–774 (2020). https://doi.org/10.1007/s00493-020-4223-9

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Mathematics Subject Classification (2010)

  • 05C15