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Towards Tight Bounds for Testing k-Colorability

Authors Nick Kushnir, Asaf Shapira

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LIPIcs.APPROX-RANDOM.2026.35.pdf
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Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • k-colorability
  • property testing
  • sample complexity
  • graph property testing
  • random graphs

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Abstract

Determining the sample complexity for testing k-colorability is perhaps the most well studied problem in property testing. It was (implicitly) introduced almost 50 years ago by Bollobás, Erdős, Simonovits and Szemerédi, who used the regularity lemma in order to give a tower-type bound for this problem. This bound has been successively improved in a long list of works, bringing the state-of-the-art bounds to lie between Ω(1/ε) and O((k/ε)log²(1/ε)). We obtain the following new results:  
ii) Our first main result is an improved O((k/ε)log(1/ε)) upper bound, bringing the sample complexity closer to "truly" linear in ε. To prove this result, we improve upon a variant of the container method, introduced recently in the breakthrough paper of Blais and Seth. 
iii) Perhaps the most important gap in our understanding of this problem is that while the best known upper bound increases with k, the lower bound is independent of k. Our second main result fills this gap by providing a new Ω((log k)/ε) lower bound, improving upon a result of Alon and Krivelevich from 2002. We conjecture that this is the true sample complexity of k-colorability.

Cite As Get BibTex

Nick Kushnir and Asaf Shapira. Towards Tight Bounds for Testing k-Colorability. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.35

Author Details

Nick Kushnir
  • School of Mathematics, Tel Aviv University, Israel
Asaf Shapira
  • School of Mathematics, Tel Aviv University, Israel

Funding

  • Shapira, Asaf: Supported in part by ERC Consolidator Grant 863438.

References

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