Even Cycles in Directed Graphs
Fan Chung, Wayne Goddard, Daniel J. Kleitman · SIAM Journal on Discrete Mathematics · 1994
It is proved that every strongly connected directed graph with n nodes and at least $\lfloor ( n + 1 )^2 /4 \rfloor $ edges must contain an even cycle. This is best possible, and the structure of extremal graphs is discussed.