Covering Complete r-Graphs with Spanning Complete r-Partite r-Graphs

Abstract

An r-cut of the complete r-uniform hypergraph Krn is obtained by partitioning its vertex set into r parts and taking all edges that meet every part in exactly one vertex. In other words it is the edge set of a spanning complete r-partite subhypergraph of Krn. An r-cut cover is a collection of r-cuts such that each edge of Krn is in at least one of the cuts. While in the graph case r = 2 any 2-cut cover on average covers each edge at least 2-o(1) times, when r is odd we exhibit an r-cut cover in which each edge is covered exactly once. When r is even no such decomposition can exist, but we can bound the average number of times an edge is cut in an r-cut cover between $1+\frac1{r+1}$ and $1+\frac{1+o(1)}{\log r}$. The upper bound construction can be reformulated in terms of a natural polyhedral problem or as a probability problem, and we solve the latter asymptotically.