Coloring the power graph of a semigroup

TL;DR

This paper investigates the chromatic number of the power graph of a semigroup, proving it is at most countable, thus addressing a recent open question in algebraic graph theory.

Contribution

It establishes that the chromatic number of the power graph of any semigroup is at most countable, providing a significant answer to an open problem.

Findings

Chromatic number of power graphs of semigroups is at most countable.

Addresses and resolves a recent open question in the field.

Provides insights into the coloring properties of algebraic structures.

Abstract

Let $G$ be a semigroup. The vertices of the power graph $\mathcal{P}(G)$ are the elements of $G$, and two elements are adjacent if and only if one of them is a power of the other. We show that the chromatic number of $\mathcal{P}(G)$ is at most countable, answering a recent question of Aalipour et al.