Distance multi-labelling of graphs with weighted vertices

The channel assignment problem (CAP) which finds an efficient assignment of channels to the transmitters of a wireless network is applicable to cellular mobile system (CMS). There are lots of results on CAP for CMS where the channel separation constraint is represented by a symmetric matrix or a graph. In particular, the Philadelphia instances and the 49-cell system are benchmark CAP for CMS and are treated in many papers. The distance multi-labelling (DM) problem on a graph with weighted vertices is an effective mathematical model of CAP for CMS in which a CMS is represented by a graph with weighted vertices, where a vertex corresponds to a cell with the number of calls on it as its weight. A DM on a graph with weighted vertices is an assignment of a set of non-negative numbers to each vertex. These numbers, which are called labels, represent the channels assigned to the demand calls in each cell. DM is a generalisation of both distance labelling and graph multi-colouring. In this study, the author introduce a new method called the layering method to find a DM on a graph with weighted vertices. Using this method, we obtain optimal DM for two Philadelphia instances. For each of them, the authors obtain a DM according to the range of separation conditions, and it includes known optimal results which are individually obtained under one separation condition.