Packing chromatic number under local changes in a graph

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. It is proved that in the class of subcubic graphs the packing chromatic number is bigger than 13, thus answering an open problem from Gastineau and Togni (2016). In addition, the packing chromatic number is investigated with respect to several local operations. In particular, if Se(G) is the graph obtained from a graph G by subdividing its edge e, then χρ(G)∕2+1≤χρ(Se(G))≤χρ(G)+1.