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\v{S}olt\'es' hypergraphs

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arxiv 2406.01504 v1 pith:LFLKKWOY submitted 2024-06-03 math.CO

classification math.CO
keywords graphshypergraphshypergraphcalledchangedefinitiondeletingdetermine
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abstract

More than $30$ years ago, \v{S}olt\'es observed that the total distance of the graph $C_{11}$ does not change by deleting a vertex, and wondered about the existence of other such graphs, called \v{S}olt\'es graphs. We extend the definition of \v{S}olt\'es' graphs to \v{S}olt\'es' hypergraphs, determine all orders for which a \v{S}olt\'es' hypergraph exists, observe infinitely many uniform \v{S}olt\'es' hypergraphs, and find the \v{S}olt\'es' hypergraph with minimum size (spoiler: it is not $C_{11}$).

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  1. 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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