A note on the packing chromatic number of lexicographic products

Abstract

The packing chromatic number χρ(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(H) denotes the independence number of H.