Toward Wojda's conjecture on digraph packing

Abstract

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 $m\geq\sqrt{8n}+418275$.