Distinguishing Labellings of Cartesian Powers and Wreath Product Actions
Salihu Lawan Aliyu, Colman Albert Wizha, Bilkisu Tijjani, Tahir Umar Gaji · Asian Research Journal of Mathematics · 2026
The distinguishing number is an important invariant used to measure the extent to which symmetries of graphs and permutation group actions can be broken by vertex labelings. In this paper, we investigate distinguishing labelings arising from permutation group actions with particular emphasis on Cartesian power constructions and wreath product actions. We establish structural bounds for distinguishing numbers in terms of orbit structure, stabilizers, and base size of permutation groups. Furthermore, we analyze the behavior of distinguishing numbers under Cartesian powers of sets and derive bounds for wreath product actions of the form G ≀ Sm acting on Xm.