On the Packing Chromatic Number of Trees, Cartesian Products and Some Infinite Graphs

A subset A of vertices in a graph G is an r-packing if the distance between every pair of distinct vertices in A is more than r. The packing chromatic number, χ ρ ( G ) , is the smallest k for which there exists some partition V 1 , V 2 , … , V k of the vertex set of G such that V r is an r-packing. We obtain lower and upper bounds for the packing chromatic number of Cartesian products and subdivisions of finite graphs and study the existence of monotone colorings in trees. The infinite, planar triangular lattice and the three dimensional square lattice are shown to have unbounded packing chromatic number.