Random Channel Assignment in the Plane

In the model for random radio channel assignment considered here, points corresponding to transmitters are thrown down independently at random in the plane, and we must assign a radio channel to each point but avoid interference. In the most basic version of the model, we assume that there is a threshold d such that, in order to avoid interference, points within distance less than d must be assigned distinct channels. Thus we wish to color the nodes of a corresponding scaled unit disk graph. We consider the first n random points, and we are interested in particular in the behavior of the ratio of the chromatic number to the clique number. We show that, as n → ∞, in probability this ratio tends to 1 in the "sparse" case [when d = d(n) is such that the average degree grows more slowly than ln n] and tends to 2√3/π ∼ 1.103 in the "dense" case (when the average degree grows faster than ln n). We also consider related graph invariants, and the more general channel assignment model when assignments must satisfy "frequency-distance" constraints. © 2003 Wiley Periodicals, Inc.