Towards the essence of \v{S}olt\'es' problem

Stijn Cambie

Towards the essence of \v{S}olt\'es' problem

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arxiv 2406.03451 v1 pith:PMCZ3BNF submitted 2024-06-05 math.CO

Stijn Cambie This is my paper · ORCID

classification math.CO

keywords problemaskingconjecturesconstantdecreasesdistancedoubtsessence

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abstract

We explore the question asking for graphs $G$ for which the total distance decreases, possibly by a fixed constant $k$, upon the removal of any of its vertices. We obtain results leading to intuition and doubts for the \v{S}olt\'es' problem ($k=0$) and its conjectures.

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Uniform \v{S}olt\'es' hypergraphs and \v{S}olt\'es' weighted graphs
math.CO 2025-06 conditional novelty 7.0 of 10

Uniform Šoltés' hypergraphs first occur at order 10, exist for every order n >= 10 and most uniformities k >= 4, a non-regular 9-uniform example exists, and infinitely many weighted Šoltés' graphs exist.

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