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

TL;DR

The paper determines the packing chromatic number $\chi_\rho$ for unitary Cayley graphs of $\mathbb{Z}_n$ and develops practical algorithms to approximate it. It combines exact, factorization-based results with two metaheuristics—Local Search and Genetic Algorithm—to compute packing colorings, comparing them against a Greedy method. The authors derive closed-form expressions for $\chi_\rho(G_{Z_n})$ depending on the prime factorization of $n$, and show that GA and LS can efficiently produce high-quality colorings, with GA showing strengths on larger, more complex graphs. The work provides both theoretical characterization for a natural graph class and scalable heuristics applicable to frequency assignment and network design problems where packing colorings are relevant.

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.

Figures (1)

• Figure 1: Graph BN16

Theorems & Definitions (9)

• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Corollary 2.4
• Proposition 2.5
• Theorem 2.6
• proof
• Theorem 2.7
• proof