An directed graph is -diregular if every vertex has indegree and outdegree at least .

**Conjecture**For , every -diregular oriented graph on at most vertices has a Hamilton cycle.

The disjoint union of two regular tournaments on vertices shows that this would be best possible. For -diregular oriented graphs with an arbitrary order of vertices, Jackson conjectured the existence of a long cycle.

Kühn and Osthus [KO] conjectured that it may actually be possible to increase the size of the graph even further if we assume that the graph is strongly 2-connected.

**Problem**Is it true that for each , every -regular strongly -connected oriented graph on at most vertices has a Hamilton cycle?

## Bibliography

*[J] B. Jackson. Long paths and cycles in oriented graphs, J. Graph Theory 5 (1981), 145-157.

[KO] D. Osthus and D. Kühn, A survey on Hamilton cycles in directed graphs, European J. Combinatorics 33 (2012), 750-766.

* indicates original appearance(s) of problem.