### Abstract

We prove that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard. The best known algorithm [20] colors such a graph using O(n^{1/5}) colors. Our result immediately implies that for any constants k > 2 and c_{2} > c_{1} > 1, coloring a k-uniform c_{1}-colorable hypergraph with c_{2} colors is NP-hard; leaving completely open only the k = 2 graph case. We are the first to obtain a 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 language | English (US) |
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Title of host publication | Annual Symposium on Foundations of Computer Science - Proceedings |

Editors | D.C. Martin |

Pages | 33-40 |

Number of pages | 8 |

State | Published - 2002 |

Event | The 34rd Annual IEEE Symposium on Foundations of Computer Science - Vancouver, BC, Canada Duration: Nov 16 2002 → Nov 19 2002 |

### Other

Other | The 34rd Annual IEEE Symposium on Foundations of Computer Science |
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Country | Canada |

City | Vancouver, BC |

Period | 11/16/02 → 11/19/02 |

### Fingerprint

### ASJC Scopus subject areas

- Hardware and Architecture

### Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*(pp. 33-40)

**The hardness of 3-uniform hypergraph coloring.** / Dinur, Irit; Regev, Oded; Smyth, Clifford.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Annual Symposium on Foundations of Computer Science - Proceedings.*pp. 33-40, The 34rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, BC, Canada, 11/16/02.

}

TY - GEN

T1 - The hardness of 3-uniform hypergraph coloring

AU - Dinur, Irit

AU - Regev, Oded

AU - Smyth, Clifford

PY - 2002

Y1 - 2002

N2 - We prove that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard. The best known algorithm [20] colors such a graph using O(n1/5) colors. Our result immediately implies that for any constants k > 2 and c2 > c1 > 1, coloring a k-uniform c1-colorable hypergraph with c2 colors is NP-hard; leaving completely open only the k = 2 graph case. We are the first to obtain a 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.

AB - We prove that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard. The best known algorithm [20] colors such a graph using O(n1/5) colors. Our result immediately implies that for any constants k > 2 and c2 > c1 > 1, coloring a k-uniform c1-colorable hypergraph with c2 colors is NP-hard; leaving completely open only the k = 2 graph case. We are the first to obtain a 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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M3 - Conference contribution

SP - 33

EP - 40

BT - Annual Symposium on Foundations of Computer Science - Proceedings

A2 - Martin, D.C.

ER -