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

An r-cut of the complete r-uniform hypergraph Kr n 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 Kr n. An r-cut cover is a collection of r-cuts so that each edge of K r n 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 + 1 r+1 and 1 + 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. ∗Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, [email protected]. Research supported by a startup grant from the Department of Mathematical Sciences of the University of Delaware. †Department of Mathematics, California State University San Marcos, San Marcos, CA 92096, [email protected]. ‡Department of Mathematics, University of California San Diego, La Jolla, CA 92093, [email protected]. §Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, [email protected].