Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Blow-ups and extensions of trees in tournaments

Pierre Aboulker, Frédéric Havet, William Lochet, Raul Lopes, Lucas Picasarri-Arrieta, Clément Rambaud

Abstract

A class of acyclic digraphs $\mathscr{C}$ is linearly unavoidable if there exists a constant $c$ such that every digraph $D\in \mathscr{C}$ is contained in all tournaments of order $c\cdot |V(D)|$. The class of all acyclic digraphs is not linearly avoidable, and Fox, He, and Widgerson recently showed that this is not even the case for acyclic digraphs with bounded maximum degree. On the positive side, Thomason and Häggkvist proved that the class of oriented trees is linearly unavoidable. In this work, we generalize this result to acyclic digraphs obtained from an oriented tree by adding at most $k$ vertices, and $k$-blow-ups of oriented trees, for every fixed integer $k$. More precisely, we show that if $D$ is obtained from an oriented tree $F$ of order $n$ by adding $k$ universal vertices, then $D$ is contained in every tournament of order $2\cdot 3^{(k+1)(2k+1)} \cdot n$; and if $D$ is obtained from $F$ by replacing each vertex $u$ by an independent set $X_u$ of size $k$ and every arc $uv$ by all possible arcs from $X_u$ to $X_v$, then $D$ is contained in every tournament of order $2^{10+18k}k \cdot n$.

Blow-ups and extensions of trees in tournaments

Abstract

A class of acyclic digraphs is linearly unavoidable if there exists a constant such that every digraph is contained in all tournaments of order . The class of all acyclic digraphs is not linearly avoidable, and Fox, He, and Widgerson recently showed that this is not even the case for acyclic digraphs with bounded maximum degree. On the positive side, Thomason and Häggkvist proved that the class of oriented trees is linearly unavoidable. In this work, we generalize this result to acyclic digraphs obtained from an oriented tree by adding at most vertices, and -blow-ups of oriented trees, for every fixed integer . More precisely, we show that if is obtained from an oriented tree of order by adding universal vertices, then is contained in every tournament of order ; and if is obtained from by replacing each vertex by an independent set of size and every arc by all possible arcs from to , then is contained in every tournament of order .

Paper Structure

This paper contains 10 sections, 23 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of tournaments
  4. Median order
  5. A Kővári-Sós-Turán-like lemma
  6. Blow-ups of oriented trees
  7. Extensions of oriented trees
  8. Particular extensions of some particular forests
  9. Twin extensions of arcless digraphs
  10. 1-extensions of directed paths

Key Result

Proposition 1

Let $D$ be an acyclic digraph. For every source or sink $x$ of $D$,

Figures (1)

  • Figure 1: Illustration of the proof of Theorem \ref{['thm:1-extension_of_trees']}. The red vertices are the image of $V(F-Z)$ under $\varphi$. The green vertices represent an embedding of $F\langle Z \rangle$ in the set $A \cap N^-_T(\varphi(v)) \setminus \varphi(V(F-Z))$, which is dashed in the figure.

Theorems & Definitions (43)

  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 3: Rédei Rede34
  • Conjecture 4: Sumner
  • Theorem 5: Kühn, Mycroft, and Osthus kuhnJCTB101
  • Conjecture 7
  • Conjecture 9
  • Conjecture 10
  • Conjecture 11
  • ...and 33 more