Symmetry breaking indices for the Cartesian product of graphs
Abstract
A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing number of a graph is the minimum number of colors required for such a coloring. The distinguishing threshold of a graph $G$ is the minimum number of colors $k$ required that any arbitrary $k$coloring of $G$ is distinguishing. We prove a statement that gives a necessary and sufficient condition for a vertex coloring of the Cartesian product to be distinguishing. Then we use it to calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of nonequivalent distinguishing colorings of grids.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.00635
 Bibcode:
 2021arXiv210800635A
 Keywords:

 Mathematics  Combinatorics;
 05C09;
 05C15;
 05C76
 EPrint:
 11 pages, 4 figures