Packing chromatic number of unitary Cayley graphs of $\Bbb Z_n$ and algorithmic approaches to it

TL;DR

This paper computes the packing chromatic number of unitary Cayley graphs of cz_n and introduces two metaheuristic algorithms to approximate it, advancing understanding and computational methods for this graph parameter.

Contribution

It provides the first explicit calculation of the packing chromatic number for these graphs and proposes new algorithmic approaches for estimating it.

Findings

Exact packing chromatic number for cz_n computed

Two metaheuristic algorithms proposed for approximation

Results demonstrate effectiveness of algorithms in estimating the number

Abstract

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into $k$ disjoint non-empty classes $V_1, \dots, V_k$, such that if $u,v \in V_i$, $i\in [k]$, $u\ne v$, then the distance between $u$ and $v$ is greater than $i$. The packing chromatic number of $G$ is the smallest integer $k$ which admits a packing $k$-coloring of $G$. In this paper, the packing chromatic number of the unitary Cayley graph of $\mathbb{Z}_n$ is computed. Two metaheuristic algorithms for calculating the packing chromatic number are also proposed.