Graph labellings with variable weights, a survey

Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings as functions of such parameters has attracted substantial attention from researchers, leading to the introduction of real number graph labellings and λ-graphs. We survey recent results obtained in this area. The concept of real number graph labellings was introduced a few years ago, and in the sequel, a more general concept of λ-graphs appeared. Though the two concepts are quite new, they are so natural that there are already many results on each. In fact, even some older results fall in this area, but their authors used a different mathematical language to state their achievements. Since many of these results are so recent that they are just appearing in various journals, we would like to offer the reader a single reference for the state of art as well as to draw attention to some older results that fall in this area. Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA. E-mail: [email protected]. Research supported in part by NSF grant DMS–0072187. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské náměst́ı 25, 118 00 Prague, Czech Republic. E-mail: [email protected]. The paper was written while the author was a Fulbright scholar at the School of Mathematics, Georgia Institute of Technology, 686 Cherry St, Atlanta, GA 30332-0160.