Hunting for Directed 2-Spiders

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.

💡 Research Summary

The paper addresses a natural extremal problem in directed graphs: determining the smallest out‑degree condition that forces the appearance of a (2, ℓ)‑spider, i.e., a 1‑subdivision of an in‑star with ℓ leaves. Earlier work by Hons, Klimošová, Kucheriya, Mikšaník, Tkadlec, and Tyomkyn (2025) showed that a minimum out‑degree of 3.23·ℓ guarantees such a spider, and they conjectured that the optimal bound should be 2ℓ. The authors of the present note confirm this conjecture by proving that every digraph with minimum out‑degree at least 2ℓ contains a (2, ℓ)‑spider as a subgraph, and they demonstrate that the bound is tight because the complete digraph on 2ℓ vertices does not contain a (2, ℓ)‑spider.

The proof hinges on the notion of “extenders”. For a fixed root vertex r, a vertex x is called an i‑extender for r if there are at least i distinct vertices y such that either x→y→r or y→x→r forms a simple directed 2‑path ending at r. An i‑extender can be used to attach x to a spider rooted at r with at least i leaves. Lemma 3 (proved by induction) shows that if we have a set F of f vertices, each being a (f + 2s + i − 1)‑extender for r, and we already have a (2, s)‑spider rooted at r disjoint from F, then we can extend it to a (2, f + s)‑spider. A corollary simplifies the condition: if we have f strong extenders (each being a (2ℓ − 1)‑extender) together with a (2, ℓ − f)‑spider, we obtain a (2, ℓ)‑spider.

The main construction proceeds as follows. Let d = 2ℓ and let G be a digraph with minimum out‑degree exactly d (any digraph with larger minimum out‑degree contains such a subgraph). Partition the vertex set into A (vertices with indegree at least 2ℓ) and B (the rest). For each r∈A define A_r as the set of in‑neighbors of r that lie in A, and V_B(r) as the set of directed 2‑paths whose middle vertex lies in B and whose terminal vertex is r. Choose r∈A maximizing the quantity d·|A_r| + |V_B(r)|; averaging shows this quantity is at least d² − d. Consequently, r has many strong extenders: every vertex in A_r is a strong extender, and there are additional strong extenders C_r among the remaining vertices.

Next, delete r together with all strong extenders (A_r∪C_r) and consider the induced subgraph H on the remaining vertices. Add an undirected edge {x, y} to H whenever x→y→r or y→x→r is a 2‑path belonging to V_B(r) and avoiding the deleted vertices. Because none of the remaining vertices is a strong extender, each vertex in H has degree at most 2ℓ − 2. By Vizing’s theorem, H can be edge‑colored with 2ℓ − 1 colors. The largest color class T_r contains at least |V_B(r)|/(2ℓ − 1) edges, and each edge corresponds to a disjoint 2‑path ending at r. Extending each such 2‑path yields a (2, s)‑spider S_r rooted at r, where s ≥ |T_r|.

If s ≥ ℓ we are done. Otherwise, the number of strong extenders a + c together with s satisfies a + c + s ≥ ℓ. Selecting exactly ℓ − s of the strong extenders (forming a set F) and applying Corollary 4 to F and S_r produces a (2, ℓ)‑spider. Thus any digraph with minimum out‑degree 2ℓ contains the desired spider, establishing the optimal bound.

The authors also observe that the proof yields an almost‑linear‑time algorithm. One can first trim the graph to have exact out‑degree d in O(nℓ) time, compute the maximizing root r in the same bound, construct the sets A_r and C_r, build the auxiliary graph H, and color its edges using recent near‑linear‑time edge‑coloring algorithms (e.g., Assadi et al., 2025/2026). The final application of Lemma 3 is trivial, giving an overall time of m^{1+o(1)}.

In the concluding section, the paper discusses open problems. It suggests classifying extremal constructions that meet the 2ℓ threshold without a (2, ℓ)‑spider, extending the result to general (k, ℓ)‑spiders (the full Giant Spider Conjecture), and improving the bound for oriented graphs (where both indegree and outdegree are constrained). The authors conjecture that for oriented graphs a bound of √2·ℓ might suffice, but achieving this would likely require new techniques beyond those used here.