### Abstract

We show that if a collection of hypergraphs (1) is uniform (every edge contains exactly k vertices, for some fixed k), (2) has minimum degree asymptotic to the maximum degree, and (3) has maximum codegree (the number of edges containing a pair of vertices) asymptotically negligible compared with the maximum degree, then the chromatic index is asymptotic to the maximum degree. This means that the edges can be partitioned into packings (or matchings), almost all of which are almost perfect. We also show that the edges can be partitioned into coverings, almost all of which are almost perfect. The result strengthens and generalizes a result due to Frankl and Rödl concerning the existence of a single almost perfect packing or covering under similar circumstances. In particular, it shows that the chromatic index of a Steiner triple-system on n points is asymptotic to n 2, resolving a long-standing conjecture.

Original language | English (US) |
---|---|

Pages (from-to) | 24-42 |

Number of pages | 19 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - 1989 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*51*(1), 24-42. https://doi.org/10.1016/0097-3165(89)90074-5

**Asymptotic behavior of the chromatic index for hypergraphs.** / Pippenger, Nicholas; Spencer, Joel.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 51, no. 1, pp. 24-42. https://doi.org/10.1016/0097-3165(89)90074-5

}

TY - JOUR

T1 - Asymptotic behavior of the chromatic index for hypergraphs

AU - Pippenger, Nicholas

AU - Spencer, Joel

PY - 1989

Y1 - 1989

N2 - We show that if a collection of hypergraphs (1) is uniform (every edge contains exactly k vertices, for some fixed k), (2) has minimum degree asymptotic to the maximum degree, and (3) has maximum codegree (the number of edges containing a pair of vertices) asymptotically negligible compared with the maximum degree, then the chromatic index is asymptotic to the maximum degree. This means that the edges can be partitioned into packings (or matchings), almost all of which are almost perfect. We also show that the edges can be partitioned into coverings, almost all of which are almost perfect. The result strengthens and generalizes a result due to Frankl and Rödl concerning the existence of a single almost perfect packing or covering under similar circumstances. In particular, it shows that the chromatic index of a Steiner triple-system on n points is asymptotic to n 2, resolving a long-standing conjecture.

AB - We show that if a collection of hypergraphs (1) is uniform (every edge contains exactly k vertices, for some fixed k), (2) has minimum degree asymptotic to the maximum degree, and (3) has maximum codegree (the number of edges containing a pair of vertices) asymptotically negligible compared with the maximum degree, then the chromatic index is asymptotic to the maximum degree. This means that the edges can be partitioned into packings (or matchings), almost all of which are almost perfect. We also show that the edges can be partitioned into coverings, almost all of which are almost perfect. The result strengthens and generalizes a result due to Frankl and Rödl concerning the existence of a single almost perfect packing or covering under similar circumstances. In particular, it shows that the chromatic index of a Steiner triple-system on n points is asymptotic to n 2, resolving a long-standing conjecture.

UR - http://www.scopus.com/inward/record.url?scp=0002840320&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002840320&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(89)90074-5

DO - 10.1016/0097-3165(89)90074-5

M3 - Article

VL - 51

SP - 24

EP - 42

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -