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A counterexample to the Bollobás–Riordan conjectures on sparse graph limits

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  27 January 2021

Ashwin Sah ,
Mehtaab Sawhney ,
Jonathan Tidor  and
Yufei Zhao
Ashwin Sah
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Mehtaab Sawhney
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Jonathan Tidor
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Yufei Zhao*
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
*Corresponding author. Email: yufeiz@mit.edu
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Abstract

Bollobás and Riordan, in their paper ‘Metrics for sparse graphs’, proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom C4-counts), but with no limit expressible as a kernel.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 796 - 799
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Supported by NSF Graduate Research Fellowship Program DGE-1122374.

‡

Supported by NSF Award DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship.

References

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