Uniform Šoltés' hypergraphs and Šoltés' weighted graphs
arXiv:2506.07511
Abstract
A Šoltés' hypergraph is a hypergraph for which the removal of any of its vertices does not change its total distance. We prove that every uniform Šoltés' hypergraph has order at least , there exist uniform Šoltés' hypergraphs for almost every order or uniformity, and there exist a non-regular uniform Šoltés' hypergraph. By also providing infinitely many weighted Šoltés' graphs, we conclude that Šoltés' problem can be answered positively for the most natural generalisations of graphs.
10 pages, 1 figure