Mathematical analysis demonstrates the existence of uniform Šoltés’ hypergraphs and weighted graphs, indicating positive resolutions to the Šoltés’ problem across generalized graph classes.
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 every order at least 10 and for almost every uniformity, and there exists a non-regular uniform Šoltés’ hypergraph. By also providing infinitely many weighted Šoltés’ graphs, we conclude that the Šoltés’ problem can be answered positively for most natural generalisations of graphs.
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Cambie et al. (2026) studied this question.
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