**Conjecture**If is a -regular directed graph with no parallel arcs, then contains a collection of arc-disjoint directed cycles.

If true, would be best possible as shown by the complete symmetric digraph.

Alon et al. [AMM] showed that a -regular directed graph with no parallel arcs contains at least arc-disjoint directed cycles. It was then improved by Alon [A] who showed that every directed graph with minimum outdegree at least contains at least arc-disjoint directed cycles.

## Bibliography

[A} N. Alon, Disjoint directed cycles, J. Combinatorial Theory, Ser. B, 68 (1996), 167-178.

*[AMM] N. Alon, C. McDiarmid and M. Molloy, Edge-disjoint cycles in regular directed graphs, J. Graph Theory, 22 (1996), no. 3, 231-237.

* indicates original appearance(s) of problem.