The hardness of 3-uniform hypergraph coloring

Irit Dinur, Oded Regev, Clifford Smyth

Research output: Contribution to journalArticlepeer-review


We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] colors such a graph using O(n 1/5) colors. Our result immediately implies that for any constants k ≥ 3 and c 2 > c 1 > 1, coloring a k-uniform c 1-colorable hypergraph with c 2 colors is NP-hard; the case k = 2, however, remains wide open. This is the first hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k ≥ 4 such a result has been shown by [14], who also discussed the inherent difference between the k = 3 case and k ≥ 4. Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph [19,22] and the Schrijver graph [26]. We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has 'many' non-monochromatic edges.

Original languageEnglish (US)
Pages (from-to)519-535
Number of pages17
Issue number5
StatePublished - Sep 2005

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics


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