Unavoidable subgraphs in digraphs with large out-degrees

TL;DR

Problem: which oriented trees $T$ must occur as subgraphs in every finite digraph with large minimum out-degree $\delta^+(G)$? Approach: introduce a height-based grounding condition on $T$, prove $\delta^+$-enforcible trees must be grounded via a level-digraph construction, and develop a δ^+-enforcible family $T(k,\ell)$ built from $B^+_{k,\ell}$ with subdivisions $S^-_{k,\ell}$, plus a general embedding framework using a $\mathcal P$ property. Key results: (i) grounding is necessary for $\delta^+$-enforcible trees, (ii) for large $\delta^+(G)$ one can embed $T(k,\ell)$ under structural assumptions on the minimal subtree containing $U(T)$, (iii) a linear bound $\delta^+(G)\ge (1+\sqrt{5})\ell$ guarantees a copy of $S^-_{2,\ell}$ (giant spiders). Significance: advances the theory of fixed-subgraph containment in digraphs with large out-degree and provides structural tools potentially extending Thomassé-type conjectures.

Abstract

We ask the question, which oriented trees $T$ must be contained as subgraphs in every finite directed graph of sufficiently large minimum out-degree. We formulate the following simple condition: all vertices in $T$ of in-degree at least $2$ must be on the same 'level' in the natural height function of $T$. We prove this condition to be necessary and conjecture it to be sufficient. In support of our conjecture, we prove it for a fairly general class of trees. An essential tool in the latter proof, and a question interesting in its own right, is finding large subdivided in-stars in a directed graph of large minimum out-degree. We conjecture that any digraph and oriented graph of minimum out-degree at least $k\ell$ and $k\ell/2$, respectively, contains the $(k-1)$-subdivision of the in-star with $\ell$ leaves as a subgraph; this would be tight and generalizes a conjecture of Thomassé. We prove this for digraphs and $k=2$ up to a factor of less than $2$.

Figures (2)

• Figure 1: An example of a tree $T$ that is $\delta^+$-enforcible by Theorem \ref{['thm:groundedclass']}, highlighting the minimal subtree containing the set $U(T)$.
• Figure 2: The level digraph $G_{3,2}$.

Theorems & Definitions (14)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Conjecture 1.4: KAMAK tree conjecture
• Theorem 1.5
• Corollary 1.6
• Conjecture 1.7: Giant spider conjecture
• Theorem 1.8
• Definition 2.1
• Lemma 3.1