The number of distinguishing colorings of a Cartesian product graph

TL;DR

This paper calculates the distinguishing threshold for Cartesian product graphs and counts the number of non-equivalent distinguishing colorings of grids, advancing understanding of graph symmetries and colorings.

Contribution

It introduces formulas for the distinguishing threshold of Cartesian product graphs and enumerates non-equivalent colorings of grid graphs, providing new insights into graph automorphisms.

Findings

Derived the distinguishing threshold for Cartesian product graphs.

Counted the number of non-equivalent distinguishing colorings of grid graphs.

Abstract

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold $\theta(G)$ of a graph $G$ is the minimum number of colors $k$ required that any arbitrary $k$-coloring of $G$ is distinguishing. In this paper, we calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of grids.