The structure of $\Delta(1, 2, 2)$-free tournaments

TL;DR

This paper characterizes the structure of tournaments excluding a specific five-vertex subtournament, providing a comprehensive description and bounds on chromatic number, transitive subtournaments, and cyclic triangles, with implications for tournament theory.

Contribution

It extends the structural understanding of tournaments by classifying those excluding the $ ext{Delta}(1, 2, 2)$ configuration, introducing new operations and bounds.

Findings

Tournaments excluding $ ext{Delta}(1, 2, 2)$ are either small, derived from transitive tournaments, or obtained via specific operations.

Provides tight bounds on chromatic number, largest transitive subtournament, and cyclic triangles for these tournaments.

The bounds are proven to be optimal.

Abstract

We extend the list of tournaments $S$ for which the complete structural description for tournaments excluding $S$ as a subtournament is known. Specifically, let $\Delta(1, 2, 2)$ be a tournament on five vertices obtained from a cyclic triangle by substituting a two-vertex tournament for two of its vertices. In this paper, we show that tournaments excluding $\Delta(1, 2, 2)$ as a subtournament are either isomorphic to one of three small tournaments, obtained from a transitive tournament by reversing edges in vertex-disjoint directed paths, or obtained from a smaller tournament with the same property by applying one of two operations. In particular, one of these operations creates a homogeneous set that induces a subtournament isomorphic to one of three fixed tournaments, and the other creates a homogeneous pair such that their union induces a subtournament isomorphic to a fixed tournament. As an application of this result, we present an upper bound for the chromatic number, a lower bound for the size of a largest transitive subtournament, and a lower bound for the number of vertex-disjoint cyclic triangles for such tournaments. The bounds that we present are all best possible.