Distinguishing invariants of Cayley Graphs, Hamming graphs, and wreath product actions

Abstract

Let $Γ$ be a nontrivial group that acts on a graph $G=(V,E)$. The total distinguishing number $D''_Γ(V\cup E)$ (distinguishing index $D'_Γ(E)$) of the action is the least number of labels required for a total labeling (edge labeling) of $G$ so that the only elements of $Γ$ that preserve the labeling belong to the pointwise stabilizer of $V\cup E$ (pointwise stabilizer of $E$). We obtain upper bounds of these invariants for Cayley graphs and under various group actions, including free actions, wreath product actions, and coordinatewise actions on Cartesian powers of graphs. Our first main result shows that, if $Γ$ acts faithfully on $G$, with $|V|\geq 2, |E|\geq 1$ and the natural action of the wreath product $W=Γ\wr S_k$ on $V^k$ induces an action of $W$ on the Cartesian power $G^{k}=(V^{k},E_k)$, then $ D''_{W}(V^k\cup E_k)\le D''_Γ(V\cup E)+k-1 $ for any $k\geq 2$. We also establish the sharpness of several of our bounds through examples involving cyclic, symmetric, and alternating groups. Moreover, we determine these distinguishing invariants for the symmetric group action on Hamming graphs.