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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics