Spanning trees in dense directed graphs

Abstract

In 2001, Komlós, Sárközy and Szemerédi proved that, for each $\alpha >0$, there is some $c>0$ and $n_0$ such that, if $n\ge n_0$, then every n-vertex graph with minimum degree at least $(1/2+\alpha )n$ contains a copy of every n-vertex tree with maximum degree at most $cn/\log n$. We prove the corresponding result for directed graphs. That is, for each $\alpha >0$, there is some $c>0$ and $n_0$ such that, if $n\ge n_0$, then every n-vertex directed graph with minimum semi-degree at least $(1/2+\alpha )n$ contains a copy of every n-vertex oriented tree whose underlying maximum degree is at most $cn/\log n$.

As with Komlós, Sárközy and Szemerédi's theorem, this is tight up to the value of c. Our result improves a recent result of Mycroft and Naia, which requires the oriented trees to have underlying maximum degree at most Δ, for any constant $\mathrm{\Delta }\in \mathbb{N}$ and sufficiently large n. In contrast to these results, our methods do not use Szemerédi's regularity lemma.

Introduction

Given two graphs H and G, when may we expect to find a copy of H in G? In general, this decision problem is NP-complete, and therefore we seek simple conditions on G which imply it contains a copy of H. An important early result is Dirac's theorem from 1952 that, when $n\ge 3$, any n-vertex graph with minimum degree at least $n/2$ contains a cycle through every vertex, that is, a Hamilton cycle. This is a particular instance of the following meta-question, which has seen much subsequent study. Given an n-vertex graph H, what is the lowest minimum degree condition on an n-vertex graph G which guarantees it contains a copy of H? As such a copy of H would contain every vertex in G, we say it is a spanning copy of H.

This question has been studied for many different graphs H, for example when H is a K-factor for some small fixed graph K [9], [15], the k-th power of a Hamilton cycle for any $k\ge 2$ [12] and when H has bounded chromatic number and maximum degree, and sublinear bandwith [4]. For more details on these results, and those for other graphs, see the survey by Kühn and Osthus [14]. Here, we will concentrate on the minimum degree required to guarantee different spanning trees.

Komlós, Sárközy and Szemerédi [11] proved in 1995 that, for each $\alpha ,\mathrm{\Delta }>0$, there is some $n_0$ such that, if $n\ge n_0$, then every n-vertex graph with minimum degree at least $(1/2+\alpha )n$ contains a copy of every n-vertex tree with maximum degree at most Δ, thus confirming a conjecture of Bollobás [2]. This result is furthermore notable as one of the earliest applications of the blow-up lemma. In 2001, Komlós, Sárközy and Szemerédi [13] relaxed the maximum degree condition, showing that, for each $\alpha >0$, there is some $c>0$ and $n_0$ such that, if $n\ge n_0$, then every n-vertex graph with minimum degree at least $(1/2+\alpha )n$ contains a copy of every n-vertex tree with maximum degree at most $cn/\log n$. This is tight up to the constant c. In 2010, Csaba, Levitt, Nagy-György and Szemerédi [5] showed that, in the other direction, the degree bound in the graph can be reduced for trees with constant maximum degree. That is, they showed that, for each $\mathrm{\Delta }>0$, there is some $C>0$ such that every n-vertex graph with minimum degree at least $n/2+C\log n$ contains a copy of every n-vertex tree with maximum degree at most Δ. This is tight up the constant C, and, moreover, unlike the previous results, did not use Szemerédi's regularity lemma.

In this paper, we will prove the corresponding version of the result of Komlós, Sárközy and Szemerédi [13] from 2001 for directed graphs (digraphs) instead of graphs. The minimum semidegree of a digraph D, denoted by ${\delta }^0(D)$, is the smallest in- or out-degree over the vertices in D, that is, ${\delta }^0(D)={\min }_{v\in V(D),\diamond \in \{+,-\}}{d}^{\diamond }(v)$. Ghouila-Houri [8] solved the minimum semidegree problem for the directed Hamilton cycle, showing that, if an n-vertex digraph D has ${\delta }^0(D)\ge n/2$, then it contains a directed Hamilton cycle. That is, an n-vertex cycle with the edges oriented in the same direction. DeBiasio, Kühn, Molla, Osthus and Taylor [6] showed that, when n is sufficiently large, this holds in fact for any n-vertex cycle with any orientations on its edges, except for when the edges change direction at every vertex around the cycle. This latter cycle, known as the anti-directed Hamilton cycle, is only guaranteed to appear if ${\delta }^0(D)\ge n/2+1$, as shown by DeBiasio and Molla [7].

Recently, Mycroft and Naia [18], [19] gave the first bound on the minimum semidegree required for the appearance of different spanning trees. Here, H is an oriented n-vertex tree, with some bound on the degree of its underlying (undirected) tree. Mycroft and Naia [18], [19] proved that, for each $\alpha ,\mathrm{\Delta }>0$, there is some $n_0$ such that, if $n\ge n_0$, then every n-vertex digraph with minimum semidegree at least $(1/2+\alpha )n$ contains a copy of every oriented n-vertex tree T with ${\mathrm{\Delta }}^{\pm }(T)\le \mathrm{\Delta }$. Moreover, their result holds for a slightly wider class of trees, allowing them to show that, for each $\alpha >0$, almost every labelled oriented n-vertex tree appears in every n-vertex digraph with minimum semidegree at least $(1/2+\alpha )n$.

In this paper, we introduce new methods to embed oriented trees in digraphs, relaxing the maximum degree condition to give a full directed version of Komlós, Sárközy and Szemerédi's result, as follows.

Theorem 1.1

For each $\alpha >0$, there exists $c>0$ and $n_0\in \mathbb{N}$ such that the following holds for every $n\ge n_0$. Any n-vertex digraph D with ${\delta }^0(D)\ge (1/2+\alpha )n$ contains a copy of every oriented n-vertex tree T with ${\mathit{\Delta }}^{\pm }(T)\le cn/\log n$.

We note that the undirected version follows immediately from Theorem 1.1. Indeed, given any n-vertex tree T and an n-vertex graph G, we can apply Theorem 1.1 to a copy of T with each edge oriented arbitrarily and a digraph formed from G by replacing each edge uv with an edge from u to v and an edge from v to u. This demonstrates that, as with Komlós, Sárközy and Szemerédi's result, Theorem 1.1 is tight up to the constant c. Furthermore, through Theorem 1.1 we give a new proof of the undirected result without using Szemerédi's regularity lemma, in contrast to the work of both Komlós, Sárközy and Szemerédi [11], and Mycroft and Naia [18], [19], adding to the non-regularity proof for trees with constant maximum degree by Csaba, Levitt, Nagy-György and Szemerédi [5] described above. Key to our result is to use a random embedding of part of the tree using ‘guide sets’ and embedding many leaves (and small subtrees) of the tree using ‘guide graphs’. This replaces the regularity methods of [11], [18], [19], and is sketched in Section 2, where we also outline the rest of this paper.