Hunting for Directed 2-Spiders
Grzegorz Gutowski, Gaurav Kucheriya
TL;DR
The paper proves, for every integer $l \ge 1$, that any directed graph with minimum out-degree at least $2l$ contains a $(2,l)$-spider as a subgraph, resolving the previously conjectured bound and matching a tight example given by the complete digraph on $2l$ vertices. The authors introduce extender-based techniques and a greedy extension lemma to iteratively grow a rooted $(2,l)$-spider, and they additionally provide an almost-linear-time algorithm that constructs such a spider in any suitable directed graph. The approach combines a careful partition of vertices by in-degree, an extension framework, and an edge-coloring step (via a Vizing-style bound) on an auxiliary graph to obtain large disjoint 2-paths that can be assembled into the target spider. These results settle the $k=2$ case of the Giant Spider Conjecture for directed graphs, and they lay the groundwork for extending the method to general $(k,l)$-spiders and to oriented graphs.
Abstract
Hons, Klimošová, Kucheriya, Mikšaník, 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.