Hunting for Directed 2-Spiders

Hons, Klimošová, Kucheriya, Mikšaník, Tkadlec, and Tyomkyn proved that, for every integer $\mathrm{\ell }\ge 1$, every directed graph with minimum out-degree at least $3.23\cdot \mathrm{\ell }$ contains a $(2,\mathrm{\ell })$-spider (a $1$-subdivision of the in-star with $\mathrm{\ell }$ leaves) as a subgraph. They also conjectured that the bound on the minimum out-degree can be further improved to $2\mathrm{\ell }$. In this note, we confirm their conjecture by showing that every directed graph with minimum out-degree at least $2\mathrm{\ell }$ contains a $(2,\mathrm{\ell })$-spider as a subgraph. This result is best possible, as the complete directed graph with $2\mathrm{\ell }$ vertices does not contain a $(2,\mathrm{\ell })$-spider.