Real Number Labelings for Paths and Cycles

The problem of radio channel assignments with multiple levels of interference depending on distance can be modeled using graph theory. The authors previously introduced a model of labeling by real numbers. Given a graph G, possibly infinite, and real numbers k1, k2, . . . , kp ≥ 0, a L(k1, k2, . . . , kp)-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 ki if u and v are at distance i apart. We denote by λ(G; k1, k2, · · · , kp) the infimum span over such labelings f . When p = 2 it is enough to determine λ(G; k, 1) for reals k ≥ 0, which will be a piecewise linear function. Here we present these functions when p = 2 for paths, cycles, and wheels.