The distinguishing number of complete bipartite and crown graphs

TL;DR

This paper calculates the minimum number of colours needed to uniquely identify automorphisms of complete bipartite and crown graphs, extending to large symmetric and alternating subgroups.

Contribution

It determines the distinguishing number for complete bipartite and crown graphs and bounds it for large automorphism subgroups with symmetric or alternating actions.

Findings

Distinguishing number of $K_{n,n}$ is between $n-1$ and $n+1$.

Distinguishing number of crown graphs is between $oxed{ ext{ceil}( ext{}\sqrt{n-1} ext{)}}$ and $oxed{ ext{floor}( ext{}\sqrt{n} ext{)}+1$.

Provides bounds for large automorphism groups acting transitively with symmetric or alternating groups.

Abstract

The distinguishing number of a permutation group $G\leqslant\Sym(\Omega)$ is the minimum number of colours needed to colour $\Omega$ in such a way that the only colour preserving element of $G$ is the identity. The distinguishing number of a graph is the distinguishing number of its automorphism group (as a permutation group on vertices). We determine the distinguishing number of the complete bipartite graphs $K_{n,n}$ and the crown graphs $K_{n,n}-nK_2$, as well as the distinguishing number of some `large' subgroups of their automorphism groups, that is, the subgroups that are vertex- and edge-transitive and such that the induced action on each bipart is $\Alt(n)$ or $\Sym(n)$. We show that, if $G$ is a `large' group of automorphisms of $K_{n,n}$, then $n-1\leqslant D(G) \leqslant n+1$. Similarly, if $G$ is a `large' group of automorphisms of a crown graph, then $\lceil \sqrt{n-1}\rceil \leqslant D(G)\leqslant \lfloor \sqrt{n}\rfloor+1$. \smallskip \textit{Keywords:} complete bipartite graph; crown graph; distinguishing number; symmetric group; alternating group