The Hardness of 3-Uniform Hypergraph Coloring

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.

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Correspondence to Irit Dinur*.

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* Research supported by NSF grant CCR-9987845.

† Supported by an Alon Fellowship and by NSF grant CCR-9987845.

‡ Work supported in part by NSF grants CCF-9988526 and DMS 9729992, and the State of New Jersery.

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Dinur*, I., Regev†, O. & Smyth‡, C. The Hardness of 3-Uniform Hypergraph Coloring. Combinatorica 25, 519–535 (2005).

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Mathematics Subject Classification (2000):

  • 68Q17