On the unavoidability of oriented trees

Abstract

A digraph is {\it $n$-unavoidable} if it is contained in every tournament of order $n$. We first prove that every arborescence of order $n$ with $k$ leaves is $(n+k-1)$-unavoidable. We then prove that every oriented tree of order $n$ ($n\ge 2$) with $k$ leaves is $(\frac{3}{2}n+\frac{3}{2}k-2)$-unavoidable and $(\frac{9}{2}n-\frac{5}{2}k-\frac{9}{2})$-unavoidable, and thus $(\frac{21}{8}n-\frac{47}{16})$-unavoidable. Finally, we prove that every oriented tree of order $n$ with $k$ leaves is $(n+144k^2-280k+124)$-unavoidable.