Domination in 3-tournaments

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, Gyárfás 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.

arXiv comment3 pages

References in corpus (3)

• Large Supports are required for Well-Supported Nash Equilibria
• Problem collection from the IML programme: Graphs, Hypergraphs, and Computing
• More differences than multiple sums