Distinguishing partitions of complete multipartite graphs

TL;DR

This paper explores the existence of distinguishing partitions in complete multipartite graphs, linking them to asymmetric hypergraphs and providing asymptotic results based on combinatorial counting methods.

Contribution

It establishes a connection between distinguishing partitions and asymmetric hypergraphs, and proves an asymptotic existence result for certain multipartite graphs.

Findings

Existence of distinguishing partitions is equivalent to asymmetric hypergraphs with specific edge sizes.

Asymptotic existence results are proven for graphs with small part sizes and large numbers of parts.

Counting particular classes of trees is key to estimating the number of such partitions.

Abstract

A \textit{distinguishing partition} of a group $X$ with automorphism group ${aut}(X)$ is a partition of $X$ that is fixed by no nontrivial element of ${aut}(X)$. In the event that $X$ is a complete multipartite graph with its automorphism group, the existence of a distinguishing partition is equivalent to the existence of an asymmetric hypergraph with prescribed edge sizes. An asymptotic result is proven on the existence of a distinguishing partition when $X$ is a complete multipartite graph with $m_1$ parts of size $n_1$ and $m_2$ parts of size $n_2$ for small $n_1$, $m_2$ and large $m_1$, $n_2$. A key tool in making the estimate is counting the number of trees of particular classes.