Uniform Šoltés' hypergraphs and Šoltés' weighted graphs

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 $10$, 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.

arXiv comment10 pages, 1 figure