Optimal channel assignments for lattices with conditions at distance two

The problem of radio channel assignments with multiple levels of interference can be modeled using graph theory. Given a graph G, possibly infinite, and real numbers k₁, k₂, ..., k_p ≥ 0, a L(k₁, k₂, ..., k_p)-labeling of G assigns real numbers f(x) ≥ 0 to the vertices x, such that the labels of vertices u and v differ by at least k_i if u and v are at distance i apart. We denote by λ(G; k₁, k₂, ..., k_p) the infimum span over such labelings f. We survey this new theory of real number labelings. When p - 2 it is enough to determine λ(G; k, 1) for reals k ≥ 0; which will be a piecewise linear function. We present the function for the square lattice (grid) and for the hexagonal lattice. For the triangular lattice, we have also solved it except for the range 1/2 ≤ k ≤ 4/5.