The exact minimum total degree threshold for the square of a Hamilton cycle in digraphs

The Pósa-Seymour conjecture establishes the minimum degree threshold required to guarantee the presence of the $k$th power of a Hamilton cycle in a graph. Following numerous partial results, Komlós, Sárközy, and Szemerédi confirmed the conjecture holds for all sufficiently large graphs. Treglown later conjectured the analogous minimum semi-degree threshold for forcing the $k$th power of a Hamilton cycle in a digraph. Subsequently, DeBiasio et al. proposed a conjecture on the minimum total degree threshold for the same problem. In this paper we settle the conjecture of DeBiasio et al. for $k=2$. Specifically, we prove that every sufficiently large $n$-vertex digraph with minimum total degree at least $8n\mathrm{/}5-c$ contains the square of a Hamilton cycle, where $c=2$ if $n\equiv 2,4(\mathrm{m}\mathrm{o}\mathrm{d}5)$, and $c=1$ otherwise.