Distinguishing Labellings of Cartesian Powers and Wreath Product Actions
Salihu Lawan Aliyu *
Department of Mathematics and Computer Science, Kashim Ibrahim University Maiduguri, Borno, Nigeria.
Colman Albert Wizha
Department of Mathematics, University of Maiduguri, Borno, Nigeria.
Bilkisu Tijjani
Department of Mathematical Sciences, Capital City University, Kano, Nigeria.
Tahir Umar Gaji
Department of Mathematics and Statistics, Integral University Lucknow, India.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Distinguishing number, wreath product, Cartesian power, symmetry breaking, base size