Further progress on Wojda's conjecture

Two digraphs of order $n$ are said to pack if they can be found as edge-disjoint subgraphs of the complete digraph of order $n$. It is well established that if the sum of the sizes of the two digraphs is at most $2n-2$, then they pack, with this bound being sharp. However, it is sufficient for the size of the smaller digraph to be only slightly below $n$ for the sum of their sizes to significantly exceed this threshold while still guaranteeing the existence of a packing. In 1985, Wojda conjectured that for any $2\le m\le n\mathrm{/}2$, if one digraph has size at most $n-m$ and the other has size less than $2n-\lfloor n\mathrm{/}m\rfloor$, then the two digraphs pack. It was previously known that this conjecture holds for $m=\mathrm{\Omega }(\sqrt{n})$. In this paper, we confirm it for $m\ge 93$ and $n\ge 31m$.