Hunting for Directed 2-Spiders

TL;DR

This paper proves that every directed graph with minimum out-degree at least 2 times ll contains a specific type of subgraph called a (2,ll)-spider, confirming a previous conjecture and establishing a tight bound.

Contribution

The authors confirm the conjecture that the minimum out-degree bound can be improved to 2ll for containing a (2,ll)-spider, establishing the result as optimal.

Findings

Minimum out-degree of 2ll guarantees a (2,ll)-spider.

Complete directed graph with 2ll vertices does not contain the spider.

Result confirms the conjecture and is tight.

Abstract

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