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Property B: Two-Coloring Non-Uniform Hypergraphs

Authors Jaikumar Radhakrishnan, Aravind Srinivasan



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Author Details

Jaikumar Radhakrishnan
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
Aravind Srinivasan
  • Department of Computer Science and UMIACS, University of Maryland at College Park, MD, USA

Acknowledgements

We thank the reviewers of the current and previous versions of this paper for their constructive suggestions that helped us correct some errors and improve the presentation.

Cite AsGet BibTex

Jaikumar Radhakrishnan and Aravind Srinivasan. Property B: Two-Coloring Non-Uniform Hypergraphs. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 31:1-31:8, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.FSTTCS.2021.31

Abstract

The following is a classical question of Erdős (Nordisk Matematisk Tidskrift, 1963) and of Erdős and Lovász (Colloquia Mathematica Societatis János Bolyai, vol. 10, 1975). Given a hypergraph ℱ with minimum edge-size k, what is the largest function g(k) such that if the expected number of monochromatic edges in ℱ is at most g(k) when the vertices of ℱ are colored red and blue randomly and independently, then we are guaranteed that ℱ is two-colorable? Duraj, Gutowski and Kozik (ICALP 2018) have shown that g(k) ≥ Ω(log k). On the other hand, if ℱ is k-uniform, the lower bound on g(k) is much higher: g(k) ≥ Ω(√{k / log k}) (Radhakrishnan and Srinivasan, Rand. Struct. Alg., 2000). In order to bridge this gap, we define a family of locally-almost-uniform hypergraphs, for which we show, via the randomized algorithm of Cherkashin and Kozik (Rand. Struct. Alg., 2015), that g(k) can be much higher than Ω(log k), e.g., 2^Ω(√{log k}) under suitable conditions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
Keywords
  • Hypergraph coloring
  • Propery B

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References

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