The distinguishing number of complete bipartite and crown graphs

TL;DR

This work determines the distinguishing numbers for two central bipartite graph families, $K_{n,n}$ and crown graphs $K_{n,n}-nK_2$, and analyzes $D(G)$ for large automorphism subgroups acting vertex- and edge-transitively with $(G^+)^{\Delta}$ equal to $\Alt(n)$ or $\Sym(n)$. It provides tight bounds and exact values by classifying large subgroups within $Aut(\Gamma)$, employing diagonal embeddings $\text{Diag}_{\varphi}(H\times H)$ and partitions that force trivial stabilizers. The main results show $D(G)$ is $n-1$, $n$, or $n+1$ for $K_{n,n}$ depending on the prescribed $G^+$, and $\lfloor \sqrt{n}\rfloor+1$ or $\lceil \sqrt{n-1}\rceil$ for crown graphs, with notable small-case exceptions where $D(G)=3$. These findings advance symmetry-breaking colorings in highly symmetric bipartite graphs and inform broader structural classifications of graph automorphism actions.

Abstract

The distinguishing number of a permutation group $G\leqslant\Sym(Ω)$ is the minimum number of colours needed to colour $Ω$ 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

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Proposition 2.2
• Remark 2.3
• proof
• Proposition 3.1
• proof