While the edges of every tournament can be covered with two spanning acyclic subgraphs, this is not so if we set out to cover all acyclic H-subgraphs of a tournament with spanning acyclic subgraphs, even for very simple H such as the 2-edge directed path or the 2-edge out-star. We prove new bounds for the minimum number of elements in such coverings and for some H our bounds determine the exact order of magnitude. A k-tournament is an orientation of the complete k-graph, where each k-set is given a total order (so tournaments are 2-tournaments). As opposed to tournaments, already covering the edges of a 3-tournament with the minimum number of spanning acyclic subhypergraphs is a nontrivial problem. We prove a new lower bound for this problem which asymptotically matches the known lower bound of covering all ordered triples of a set.
Bibliographical noteFunding Information:
∗Supported by Israel Science Foundation (grant No. 1082/16).
© The author. Released under the CC BY-ND license (International 4.0).
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics