**Conjecture**Every k-arc-strong tournament decomposes into k spanning strong digraphs.

Conjecture 8 implies Kelly's conjecture (*Every regular tournament of order can be decomposed into Hamilton directed cycles.*) which has been proved for tournaments of sufficiently large order by Kühn and Osthus [KO].

Bang-Jensen and Yeo [BY] gave several results supporting this conjecture. For example they proved it for -arc-strong tournaments with minimum in- and out-degree at least .

## Bibliography

*[BY] J. Bang-Jensen, A. Yeo, Decomposing k-arc-strong tournaments into strong spanning subdigraphs, Combinatorica 24 (2004) 331–349.

[KO] Daniela Kühn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146.

* indicates original appearance(s) of problem.