Every strong digraph has a spanning strong subgraph with at most \(n+2\alpha-2\) arcs (Q1405121)

The authors prove that every strong digraph \(D\) with \(n\) vertices and independence number \(\alpha\) has a spanning strong subgraph with at most \(n+2\alpha- 2\) arcs. Such a spanning subgraph can be found in polynomial time. This result implies that \(D\) is spanned by at most \(2\alpha-1\) directed cycles. In 1964 Tibor Gallai conjectured that \(\alpha\) directed cycles are enough.