Uniform \v{S}olt\'es' hypergraphs and \v{S}olt\'es' weighted graphs

TL;DR

This paper investigates ;olte9s' hypergraphs and weighted graphs, establishing existence results, minimal sizes, and properties, thus extending the classical problem to broader hypergraph and weighted graph contexts.

Contribution

It proves minimal size bounds for uniform ;olte9s' hypergraphs, demonstrates their existence for various parameters, and introduces weighted ;olte9s' graphs, broadening the problem's scope.

Findings

Every uniform ;olte9s' hypergraph has at least 10 vertices.

Existence of uniform ;olte9s' hypergraphs for almost all sizes and uniformities.

Infinitely many weighted ;olte9s' graphs are constructed.

Abstract

A \v{S}olt\'es' hypergraph is a hypergraph for which the removal of any of its vertices does not change its total distance. We prove that every uniform \v{S}olt\'es' hypergraph has order at least $10$, there exist uniform \v{S}olt\'es' hypergraphs for almost every order or uniformity, and there exist a non-regular uniform \v{S}olt\'es' hypergraph. By also providing infinitely many weighted \v{S}olt\'es' graphs, we conclude that \v{S}olt\'es' problem can be answered positively for the most natural generalisations of graphs.