Distinguishing partitions and asymmetric uniform hypergraphs

Abstract

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 m ≥ f ( n ), where f (2) = f (14) = 6, f (6) = 5, and f ( n )= ⌊ log 2 ( n + 1) ⌋ + 2 otherwise. It follows that a distinguishing partition of K m ( n ) = K n , n , ..., n , or equivalently for the wreath product action S n Wr S m , exists if and only if m ≥ f ( n ).