Structure theorem for U5-free tournaments

TL;DR

This paper characterizes prime tournaments that do not contain a specific 5-vertex subtournament $U_5$, providing a structural theorem and implications for transitive subtournaments.

Contribution

It offers a complete structural description of $U_5$-free tournaments, including a construction method and bounds on transitive subtournaments.

Findings

Prime $U_5$-free tournaments are either a specific tournament $T_n$ or have a tripartition with transitive unions.

All $U_5$-free tournaments can be generated from prime $U_5$-free tournaments.

Every $U_5$-free tournament with $n$ vertices has a transitive subtournament of size at least $n^{rac{ ext{log}_3 2}$}.

Abstract

Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2 \rightarrow v_1$, and $v_i \rightarrow v_j$ if $j-i \equiv 1$, $2 \pmod{5}$ and ${i,j} \neq {1,2}$. In this paper we describe the tournaments which do not have $U_5$ as a subtournament. Specifically, we show that if a tournament $G$ is "prime"---that is, if there is no subset $X \subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v \in V(G) \backslash X$, either $v \rightarrow x$ for all $x \in X$ or $x \rightarrow v$ for all $x \in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$ or $V(G)$ can be partitioned into sets $X$, $Y$, $Z$ such that $X \cup Y$, $Y \cup Z$, and $Z \cup X$ are transitive. From the prime $U_5$-free tournaments we can construct all the $U_5$-free tournaments. We use the theorem to show that every $U_5$-free tournament with $n$ vertices has a transitive subtournament with at least $n^{\log_3 2}$ vertices, and that this bound is tight.