# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## FREQUENCY PLANNING AND RAMIFICATIONS OF COLORING

Andreas Eisenblätter, Martin Grötschel  and  Arie M.C.A. Koster

Konrad-Zuse-Zentrum für Informationstechnik Berlin
Takustraße 7, D-14195 Berlin, Germany

## Abstract

This paper surveys frequency assignment problems coming up in planning wireless communication services. It particularly focuses on cellular mobile phone systems such as GSM, a technology that revolutionizes communication. Traditional vertex coloring provides a conceptual framework for the mathematical modeling of many frequency planning problems. This basic form, however, needs various extensions to cover technical and organizational side constraints. Among these ramifications are T-coloring and list coloring. To model all the subtleties, the techniques of integer programming have proven to be very useful.

The ability to produce good frequency plans in practice is essential for the quality of mobile phone networks. The present algorithmic solution methods employ variants of some of the traditional coloring heuristics as well as more sophisticated machinery from mathematical programming. This paper will also address this issue.

Finally, this paper discusses several practical frequency assignment problems in detail, states the associated mathematical models, and also points to public electronic libraries of frequency assignment problems from practice. The associated graphs have up to several thousand vertices and range form rather sparse to almost complete.

Keywords: frequency assignment, graph coloring.

2000 Mathematics Subject Classification: 05-02, 05C90, 90-02, 05C15.

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Received 4 January 2001
Revised 7 June 2001