The structure of $\Delta(1, 2, 2)$-free tournaments
Abstract
We extend the list of tournaments for which the complete structural description for tournaments excluding as a subtournament is known. Specifically, let 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 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.
Keywords
Cite
@article{arxiv.2511.03234,
title = {The structure of $\Delta(1, 2, 2)$-free tournaments},
author = {Seokbeom Kim and Taite LaGrange and Mathieu Rundström and Arpan Sadhukhan and Sophie Spirkl},
journal= {arXiv preprint arXiv:2511.03234},
year = {2025}
}