Antidirected trees in directed graphs

TL;DR

This paper establishes degree conditions in directed graphs that guarantee the presence of balanced antidirected trees of smaller order, extending previous results on tree containment in graphs and digraphs.

Contribution

It combines and extends existing theorems by providing new minimum and maximum degree bounds for embedding balanced antidirected trees in digraphs.

Findings

Degree conditions ensure antidirected tree containment

Vertex degree thresholds are explicitly characterized

Results apply to large, balanced antidirected trees

Abstract

The Koml\'os-S\'ark\"ozy-Szemer\'edi (KSS) theorem establishes that a certain bound on the minimum degree of a graph guarantees it contains all bounded degree trees of the same order. Recently several authors put forward variants of this result, where the tree is of smaller order than the host graph, and the host graph also obeys a maximum degree condition. Also, Kathapurkar and Montgomery extended the KSS theorem to digraphs. We bring these two directions together by establishing minimum and maximum degree bounds for digraphs that ensure the containment of oriented trees of smaller order. Our result is restricted to balanced antidirected trees of bounded degree. More precisely, we show that for every $\gamma > 0$, $c\in\mathbb{R}$, $\ell\geq 2$ sufficiently large $n$ and all $k\geq\gamma n$, the following holds for every $n$-vertex digraph $D$ and every balanced antidirected tree $T$ with $k$ arcs whose total maximum degree is bounded by $(\log n)^c$. If $D$ has a vertex of outdegree at least $(1+\gamma)(\ell -1)k$, a vertex of indegree at least $(1+\gamma)(\ell -1)k$ and minimum semidegree $\delta^0(D)\geq\left(\frac{\ell}{2\ell -1}+\gamma\right)k$, then $D$ contains $T$.