Asymmetric Colorings of Disjoint Unions of Graphs

TL;DR

This paper studies the asymmetric coloring number of disjoint unions of graphs, establishing relationships between the number of colors needed and the asymmetrical coloring of individual components, with specific cases analyzed.

Contribution

It introduces a general relationship linking the asymmetric coloring number of disjoint graphs to the coloring of single graphs, and examines specific graph classes.

Findings

Derived a formula relating disjoint union coloring to individual graphs

Analyzed asymmetric coloring for paths, stars, cycles, hypercubes

Provided insights into automorphism-preserving colorings

Abstract

The asymmetric coloring number of a graph is the minimum number of colors needed to color its vertices, so that no non-trivial automorphism preserves the color classes. We investigate the asymmetric coloring number of graphs that are disjoint unions of graphs. We will derive a general relationship between the asymmetric coloring number of disjoint copies of graphs and the number of ways to color a single copy asymmetrically, and then look at particular cases such as disjoint copies of paths, stars, cycles, and hypercubes.