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Polychromatic Coloring of Tuples in Hypergraphs

Authors Ahmad Biniaz , Jean-Lou De Carufel , Anil Maheshwari , Michiel Smid , Shakhar Smorodinsky , Miloš Stojaković

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Hypergraph Coloring
  • Polychromatic Coloring
  • Geometric Hypergraphs
  • Cover Decomposable Hypergraphs
  • Epsilon Nets

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Abstract

A hypergraph H consists of a set V of vertices and a set E of hyperedges that are subsets of V. A t-tuple of H is a subset of t vertices of V. A t-tuple k-coloring of H is a mapping of its t-tuples into k colors. A coloring is called (t,k,f)-polychromatic if each hyperedge of E that has at least f vertices contains tuples of all the k colors. Let f_H(t,k) be the minimum f such that H has a (t,k,f)-polychromatic coloring. For a family of hypergraphs ℋ let f_H(t,k) be the maximum f_H(t,k) over all hypergraphs H in H. Determining f_H(t,k) has been an active research direction in recent years. This is challenging even for t = 1. We present several new results in this direction for t ≥ 2.  
- Let H be the family of hypergraphs H that is obtained by taking any set P of points in ℝ², setting V: = P and E: = {d ∩ P: d is a disk in ℝ²}. We prove that f_ H(2,k) ≤ 3.7^k, that is, the pairs of points (2-tuples) can be k-colored such that any disk containing at least 3.7^k points has pairs of all colors. We generalize this result to points and balls in higher dimensions. 
- For the family H of hypergraphs that are defined by grid vertices and axis-parallel rectangles in the plane, we show that f_H(2,k) ≤ √{ck ln k} for some constant c. We then generalize this to higher dimensions, to other shapes, and to tuples of larger size. 
- For the family H of shrinkable hypergraphs of VC-dimension at most d we prove that f_ H(d+1,k) ≤ c^k for some constant c = c(d). Towards this bound, we obtain a result of independent interest: Every hypergraph with n vertices and with VC-dimension at most d has a (d+1)-tuple T of depth at least n/c, i.e., any hyperedge that contains T also contains n/c other vertices. 
- For the relationship between t-tuple coloring and vertex coloring in any hypergraph H we establish the inequality 1/e⋅ tk^{1/t} ≤ f_H(t,k) ≤ f_H(1,tk^{1/t}). For the special case of k = 2, referred to as the bichromatic coloring, we prove that t+1 ≤ f_H(t,2) ≤ max{f_H(1,2), t+1}; this improves upon the previous best known upper bound.
- We study the relationship between tuple coloring and epsilon nets. In particular we show that if f_H(1,k) = O(k) for a hypergraph H with n vertices, then for any 0 < ε < 1 the t-tuples of H can be partitioned into Ω((εn/t)^t) ε-t-nets. This bound is tight when t is a constant.

Cite As Get BibTex

Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid, Shakhar Smorodinsky, and Miloš Stojaković. Polychromatic Coloring of Tuples in Hypergraphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.19

Author Details

Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Jean-Lou De Carufel
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Michiel Smid
  • School of Computer Science, Carleton University, Ottawa, Canada
Shakhar Smorodinsky
  • Department of Computer Science, Ben-Gurion University of the Negev, Be'er Sheva, Israel
Miloš Stojaković
  • Department of Mathematics and Informatics, University of Novi Sad, Serbia

Funding

  • Biniaz, Ahmad: Supported in part by NSERC.
  • De Carufel, Jean-Lou: Supported in part by NSERC.
  • Maheshwari, Anil: Supported in part by NSERC.
  • Smid, Michiel: Supported in part by NSERC.
  • Smorodinsky, Shakhar: Partially supported by Grant 1065/20 from the Israel Science Foundation, by the United States – Israel Binational Science Foundation (NSF-BSF grant no. 2022792) and by ERC grant no. 882971 "GeoScape" and by the Erdős Center.
  • Stojaković, Miloš: Partly supported by Ministry of Science, Technological Development and Innovation of Republic of Serbia (Grants 451-03-137/2025-03/200125 & 451-03-136/2025-03/200125). Partly supported by Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina (Grant No. 142-451-2686/2021).

Acknowledgements

This work was initiated at the 20th Gremo’s Workshop on Open Problems, held in Wergenstein, Switzerland, in June 2023. It was then continued at the 11th Annual Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Holetown, Barbados, in March 2024. We thank the organizers and participants of both workshops. We also thank the reviewers of SoCG 2025 who verified our proofs and provided valuable feedback.

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