Toward Wojda's conjecture on digraph packing

Given a positive integer \(m\leq n/2\), Wojda conjectured in 1985 that if \(D_1\) and \(D_2\) are digraphs of order \(n\) such that \(|A(D_1)|\leq n-m\) and \(|A(D_2)|\leq 2n-\lfloor n/m\rfloor-1\) then \(D_1\) and \(D_2\) pack. The cases when \(m=1\) or \(m = n/2\) follow from known results. Here we prove the conjecture for \(m\geq\sqrt{8n}+418275\)