نمونه متن انگلیسی مقاله
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.
A tournament is an oriented complete graph. The following generalization of tournaments to higher uniformity was suggested by Gyárfás. An r-tournament is a complete r-uniform hypergraph T where each edge has a special vertex designated as its tail. We say that a vertex set X dominates T if every vertex outside X is contained in a hyperedge whose tail is in X. The domination number of T is the minimum size of such a dominating set X. Recently Gyárfás made the following two conjectures about 3-tournaments (see ).