Domination in 3-tournaments

TL;DR

This paper proves Gyárfás's conjecture that 3-tournaments can have arbitrarily large domination numbers and refutes a related conjecture about induced triples, advancing understanding of domination in hypergraph structures.

Contribution

The paper confirms the existence of 3-tournaments with arbitrarily large domination numbers and disproves a conjecture about induced triples with the same tail.

Findings

Confirmed Gyárfás's conjecture on large domination numbers.

Refuted the conjecture about induced triples with the same tail.

Provided new insights into domination properties of 3-uniform hypergraphs.

Abstract

A 3-tournament is a complete 3-uniform hypergraph where each edge has a special vertex designated as its tail. A vertex set $X$ dominates $T$ if every vertex not in $X$ is contained in an edge whose tail is in $X$. The domination number of $T$ is the minimum size of such an $X$. Generalizing well-known results about usual (graph) tournaments, Gy\'arf\'as conjectured that there are 3-tournaments with arbitrarily large domination number, and that this is not the case if any four vertices induce two triples with the same tail. In this short note we solve both problems, proving the first conjecture and refuting the second.