Trees with few leaves in tournaments

TL;DR

This paper proves new bounds on the size of tournaments needed to embed all oriented trees with a given number of leaves, improving previous results and confirming a conjecture for large enough tournaments.

Contribution

It establishes tighter bounds on tournament size for embedding trees with leaves, advancing the understanding of tree embeddings in tournaments.

Findings

Improved bound from n+O(k^2) to n+Ck for embedding trees with k leaves.

Confirmed a conjecture that (n+k-2)-vertex tournaments contain all trees with n vertices and at most k leaves for large n.

Demonstrated that the bounds are tight up to the constant C.

Abstract

We prove that there exists $C>0$ such that any $(n+Ck)$-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves, improving the previously best known bound of $n+O(k^2)$ vertices to give a result tight up to the value of $C$. Furthermore, we show that, for each $k$, there exists $n_0$, such that, whenever $n\geqslant n_0$, any $(n+k-2)$-vertex tournament contains a copy of every $n$-vertex oriented tree with at most $k$ leaves, confirming a conjecture of Dross and Havet.