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arXiv · 2511.03234

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

Seokbeom Kim·Taite LaGrange·Mathieu Rundström·Arpan Sadhukhan·Sophie Spirkl
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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.

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Seokbeom Kim, Taite LaGrange, Mathieu Rundström, Arpan Sadhukhan, Sophie Spirkl. 2025-11-05. The structure of $\Delta(1, 2, 2)$-free tournaments. https://arxiv.org/abs/2511.03234

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