Radio channel assignment on 2-dimensional lattices

Given a set V of points in the plane, a sequence d0d1dk of nonnegative numbers and an integer n, we are interested in the problem to assign integers from 0n−1 to the points in V such that if xyV are two points with Euclidean distance less than dj, then the difference between the labels of x and y is not equal to i. This question is inspired by problems occurring in the design of radio networks, where radio channels need to be assigned to transmitters in such a way that interference is minimized. In this paper we consider the case that the set of points are the points of a 2-dimensional lattice. Recent results by McDiarmid and Reed show that if only one constraint d0 is given, good labellings can be obtained by using so-called strict tilings. We extend these results to the case that higher level constraints d0d1dk occur. In particular we study conditions that guarantee that a strict tiling, satisfying only the one constraint d0, can be transformed to a strict tiling satisfying the higher order constraints as well. Special attention is devoted to the case that the points are the points of a triangular lattice.