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


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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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