Antidirected trees in directed graphs

Abstract

The Komlós-Sárközy-Szemerédi (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 $γ> 0$, $c\in\mathbb{R}$, $\ell\geq 2$ sufficiently large $n$ and all $k\geqγn$, 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 $(1+γ)(\ell -1)k$, a vertex of indegree at least $(1+γ)(\ell -1)k$ and minimum semidegree $δ^0(D)\geq\left(\frac{\ell}{2\ell -1}+γ\right)k$, then $D$ contains $T$.