Modeling the packing coloring problem of graphs

The frequency assignment problem asks for assigning frequencies to transmitters in a wireless network and includes a variety of specific subproblems. In particular, the packing coloring problem comes from the regulations concerning the assignment of broadcast frequencies to radio stations. The problem is placed on a graph-theoretical footing and modeled as the packing chromatic number χρ(G)χρ(G) of a graph G. The packing chromatic number of G is the smallest integer k   such that the vertex set V(G)V(G) can be partitioned into disjoint classes X1,…,XkX1,…,Xk, with the condition that vertices in XiXi have pairwise distance greater than i. The determination of the packing chromatic number is known to be NP-hard. This paper presents an integer linear programming model and a satisfiability test model for the packing coloring problem. The proposed models were applied for studying the packing chromatic numbers of Cartesian products of paths and cycles and for the packing chromatic numbers of some distance graphs.