Recent progress in mathematics and engineering on optimal graph labellings with distance conditions

The problem of radio channel assignments with multiple levels of interference can be modelled using graph theory. The theory of integer vertex-labellings of graphs with distance conditions has been investigated for several years now, and the authors recently introduced a new model of real number labellings that is giving deeper insight into the problems. Here we present an overview of the recent outpouring of papers in the engineering literature on such channel assignment problems, with the goal of strengthening connections to applications. Secondly, we present a new contribution to the theory, the formulas for the optimal span of labellings with conditions at distance two for finite complete bipartite graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. Research supported in part by NSF grants DMS-0072187 and DMS-0302307. An early version of this paper was presented at the CTS Conference on Combinatorics and its Applications in May, 2005, at Chiao Tung University, HsinChu, Taiwan. The first author is grateful to the CTS for its support of his travel.