Blow-ups and extensions of trees in tournaments

TL;DR

This paper extends the class of acyclic digraphs known to be linearly unavoidable in tournaments, including those formed from oriented trees with added vertices or blow-ups, providing explicit bounds for containment.

Contribution

It generalizes previous results by proving that certain complex acyclic digraphs derived from oriented trees are linearly unavoidable in tournaments, with explicit size bounds.

Findings

Acyclic digraphs from oriented trees with added universal vertices are contained in all large enough tournaments.

Blow-ups of oriented trees are also contained in all sufficiently large tournaments.

Provides explicit bounds on tournament size needed for containment of these digraphs.

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 H\"aggkvist 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 $2^{10+18k}k \cdot n$.