Directed Cycles with Two Chords and Strong Spanning Directed Subgraphs with Few Arcs

We prove the conjecture of D. A. Marcus (1981) that every strongly 2-arc-connected directed graph has a directed cycle with at least two chords. As a consequence, every strongly 2-arc-connected directed graph withmarcs has a spanning strong directed subgraph with less than 23 arcs. The constant 23 is best possible.