Cycles in directed graphs

There are many theorems concerning cycles in graphs for which it is natural to seek analogous results for directed graphs. I will survey

recent progress on certain questions of this type. New results include

(i) a solution to a question of Thomassen on an analogue of Dirac’s theorem

for oriented graphs,

(ii) a theorem on packing cyclic triangles in tournaments that “almost” answers a question of Cuckler and Yuster, and

(iii) a bound for the smallest feedback arc set in a digraph with no short directed cycles, which is optimal up to a constant factor and extends a result of Chudnovsky, Seymour and Sullivan.

These are joint work respectively with (i) Kuhn and Osthus, (ii) Sudakov, and (iii) Fox and Sudakov.