Distinguishing partitions and asymmetric uniform hypergraphs

M. N. Ellingham, Justin Z. Schroeder


A distinguishing partition for an action of a group Γ on a set X is a partition of X that is preserved by no nontrivial element of Γ. As a special case, a distinguishing partition of a graph is a partition of the vertex set that is preserved by no nontrivial automorphism. In this paper we provide a link between distinguishing partitions of complete equipartite graphs and asymmetric uniform hypergraphs. Suppose that m ≥ 1 and n ≥ 2. We show that an asymmetric n-uniform hypergraph with m edges exists if and only if mf(n), where f(2) = f(14) = 6, f(6) = 5, and f(n)= ⌊ log2(n + 1) ⌋ + 2 otherwise. It follows that a distinguishing partition of Km(n) = Kn, n, ..., n, or equivalently for the wreath product action Sn Wr Sm, exists if and only if mf(n).


complete equipartite graph, distinguishing number, distinguishing partition, asymmetric uniform hypergraph

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DOI: https://doi.org/10.26493/1855-3974.195.d00

ISSN: 1855-3974

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