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Attempting perfect hypergraphs

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Abstract

We study several extensions of the notion of perfect graphs to k-uniform hypergraphs. One main definition extends to hypergraphs the notion of perfect graphs based on coloring. Let G be a k-uniform hypergraph. A coloring of a k-uniform hypergraph G is proper if it is a coloring of the (k − 1)-tuples with elements in V(G) in such a way that no edge of G is a monochromatic \(K_k^{k - 1}\).

A k-uniform hypergraph G is Cω-perfect if for every induced subhypergraph G′ of G we have:

  • if X ⊆ V(G′) with ∣X∣ < k − 1, then there is a proper (ω(G′) − k + 2)-coloring of G′ (so (k − 1)-tuples are colored) that restricts to a proper (ω(G′) − k + 2)-coloring of lkG′(X) (so (k − ∣X∣ − 1)-tuples are colored).

Another main definition is the following: A k-uniform hypergraph G is hereditary perfect (or, briefly, H-perfect) if all links of sets of (k − 2) vertices are perfect graphs.

The notion of Cω perfectness is not closed under complementation (for k > 2) and we define G to be doubly perfect if both G and its complement are Cω perfect. We study doubly-perfect hypergraphs: In addition to perfect graphs nontrivial doubly-perfect graphs consist of a restricted interesting class of 3-uniform hypergraphs, and within this class we give a complete characterization of doubly-perfect H-perfect hypergraphs.

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Acknowledgment

We are thankful to Gabor Simonyi, Vitaly Voloshin and an anonymous referee for helpful remarks. We thank the referee especially for detecting several mistakes in an earlier version of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Maria Chudnovsky

  2. Efi Arazi School of Computer Science, Reichman University, Herzliya, 4610101, Israel

    Gil Kalai

  3. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel

    Gil Kalai

Authors
  1. Maria Chudnovsky
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  2. Gil Kalai
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Corresponding author

Correspondence to Gil Kalai.

Additional information

Dedicated to Nati Linial and his vision of high dimensional combinatorics

This material is based upon work supported in part by the U. S. Army Research Office under grant number W911NF-16-1-0404, and supported by NSF grant DMS 1763817 and ERC advanced grant 320924.

Supported by ERC advanced grant 320924 and by ISF Grant 1612/17.

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Chudnovsky, M., Kalai, G. Attempting perfect hypergraphs. Isr. J. Math. 256, 133–151 (2023). https://doi.org/10.1007/s11856-023-2506-8

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  • DOI: https://doi.org/10.1007/s11856-023-2506-8

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