Distance multi-labelling of graphs with weighted vertices

• Published in: IET communications Vol. 11; no. 16; pp. 2524 - 2530
• Main Authors: Kim, Byeong Moon, Rho, Yoomi, Song, Byung Chul
• Format: Journal Article
• Language: English
• Published: The Institution of Engineering and Technology 09.11.2017
• Subjects: cellular mobile system; channel assignment problem; channel separation constraint; distance graph multilabelling; distance multilabelling problem; graph multicolouring; graph theory; mathematical analysis; mathematical model; radio networks; radio transmitters; Research Article; telecommunication channels; transmitters; weighted vertices; wireless network; radio networks; cellular mobile system; telecommunication channels; distance multilabelling problem; transmitters; graph theory; channel separation constraint; mathematical analysis; weighted vertices; distance graph multilabelling; mathematical model; radio transmitters; wireless network; channel assignment problem; graph multicolouring
• ISSN: 1751-8628; 1751-8636; 1751-8636
• DOI: 10.1049/iet-com.2016.0708

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.