A note on the packing chromatic number of lexicographic products

TL;DR

This paper investigates bounds for the packing chromatic number of lexicographic graph products, providing conditions under which these bounds are tight, thus advancing understanding of coloring properties in complex graph structures.

Contribution

It introduces new upper and lower bounds for the packing chromatic number of lexicographic products, with conditions for their exactness.

Findings

Bounds often coincide under certain conditions

Bounds are tight when |V(H)| - α(H) ≥ diam(G) - 1

Provides insights into coloring complexities of lexicographic products

Abstract

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. In this short note we present upper and lower bound for the packing chromatic number of the lexicographic product $G\circ H$ of graphs $G$ and $H$. Both bounds coincide in many cases. In particular this happens if $|V(H)|-\alpha(H)\geq {\rm diam}(G)-1$, where $\alpha(G)$ denotes the independence number of $G$.