A note on the packing chromatic number of lexicographic products

Abstract

The packing chromatic number χ ρ (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 • H of graphs G and H. Both bounds coincide in many cases. In particular this happens if |V (H)| − α(H) ≥ diam(G) − 1, where α(G) denotes the independence number of G.