# Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament

**Conjecture**If is a tournament of order , then it contains arc-disjoint transitive subtournaments of order 3.

If true the conjecture would be tight as shown by any tournament whose vertex set can be decomposed into sets of size or and such that , and .

Let denote the transitive tournament of order 3. A *-packing* of a digraph is a set of arc-disjoint copies of subgraphs of .

Let be the minimum size of a -packing over all tournaments of order . The conjecture and its tightness say .

The best lower bound for so far is due to Kabiya and Yuster [KY] proved that .

## Bibliography

[KY] M. Kabiya and R. Yuster. Packing transitive triples in a tournament. Ann. Comb. 12 (2008), no. 3, 291–-306.

*[Y] R. Yuster. The number of edge-disjoint transitive triples in a tournament. Discrete Math. 287 (2004). no. 1-3,187--191.

* indicates original appearance(s) of problem.