On the unavoidability of oriented trees

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 $n$ ($n\geq 2$) 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

On the unavoidability of oriented trees

International audienc

Oriented trees and paths in digraphs

Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain order. In arbitrary digraphs and oriented graphs, conditions on the chromatic number, on the edge density, on the minimum outdegree and on the minimum semidegree have been proposed. In this survey, we review the known results, and highlight some open questions in the area

Trees with few leaves in tournaments

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.Comment: 22 pages, 3 figure

Trees with many leaves in tournaments

Sumner's universal tournament conjecture states that every (2n−2)-vertex tournament should contain a copy of every n-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a copy of the tree with fewer vertices in the tournament? Due to work initiated by Häggkvist and Thomason (for number of leaves) and Kühn, Mycroft and Osthus (for maximum degree), it is known that improvements can be made over Sumner's conjecture in some cases, and indeed sometimes an (n+o(n))-vertex tournament may be sufficient. In this paper, we give new results on these problems. Specifically, we show i) for every α>0, there exists n0∈N such that, whenever n⩾n0, every ((1+α)n+k)-vertex tournament contains a copy of every n-vertex oriented tree with k leaves, and ii) for every α>0, there exists c>0 and n0∈N such that, whenever n⩾n0, every (1+α)n-vertex tournament contains a copy of every n-vertex oriented tree with maximum degree Δ(T)⩽cn. Our first result gives an asymptotic form of a conjecture by Havet and Thomassé, while the second improves a result of Mycroft and Naia which applies to trees with polylogarithmic maximum degree

On the appearance of oriented trees in tournaments

We consider how large a tournament must be in order to guarantee the appearance of a given oriented tree. Sumner’s universal tournament conjecture states that every (2$n$−2)-vertex tournament should contain a copy of every n-vertex oriented tree. However, it is known that improvements can be made over Sumner’s conjecture in some cases by considering the number of leaves or maximum degree of an oriented tree. To this end, we establish the following results. (1) There exists $C$ > 0 such that any ($n$ + $Ck$)-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves. (2) For each $k$, there exists $n_0$ ∈ $\mathbb{N}$, such that, whenever $n$ ⩾ $n_0$, any ($n$+$k$ −2)-vertex tournament contains a copy of every $n$-vertex oriented tree with at most $k$ leaves. (3) For every α > 0, there exists $n_0$ ∈ $\mathbb{N}$ such that, whenever $n$ ⩾ $n_0$, any ((1+α)$n$+$k$)-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves. (4) For every α > 0, there exists $c$ > 0 and $n_0$ ∈ $\mathbb{N}$ such that, whenever $n$ ⩾ $n_0$, any (1 + α)$n$-vertex tournament contains a copy of any $n$-vertex oriented tree with maximum degree Δ($T$) ⩽ $cn$. (5) For all countably-infinite oriented graphs $H$, either (i) there is a countably-infinite tournament not containing $H$, or (ii) every countably-infinite tournament contains a spanning copy of $H$. (1) improves the previously best known bound of $n$ + $O(k^2)$. (2) confirms a conjecture of Dross and Havet. (3) provides an asymptotic form of a conjecture of Havet and Thomassé. (4) improves a result of Mycroft and Naia which applies to trees with polylogarithmic maximum degree. (5) extends the problem to the infinite setting, where we also consider sufficient conditions for the appearance of oriented graphs satisfying (i)

On the Unavoidability of Oriented Trees

International audienceA digraph is 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 with k leaves is (3 2 n + 3 2 k − 2)-unavoidable and (9 2 n − 5 2 k − 9 2)-unavoidable, and thus (21 8 (n − 1))-unavoidable. Finally, we prove that every oriented tree of order n with k leaves is (n + 144k 2 − 280k + 124)-unavoidable