Packing chromatic number of certain graphs

Abstract

The packing chromatic number Xp(G) of a graph G is the smallest integer k for which there exists a mapping II : V (G) → {1, 2, ⋯, k} such that any two vertices of color i are at distance at least i + 1. It is a frequency assignment problem used in wireless networks, which is also called broadcasting coloring. It is proved that packing coloring is NP-complete for general graphs and even for trees. In this paper, we study the packing chromatic number of comb graph, circular ladder, windmill, H-graph and uniform theta graph.