Real Number Graph Labellings with Distance Conditions

The theory of integer λ-labellings of a graph, introduced by Griggs and Yeh, seeks to model efficient channel assignments for a network of transmitters. To prevent interference, labels for nearby vertices must be separated by specified amounts ki depending on the distance i, 1 ≤ i ≤ p. Here we expand the model to allow real number labels and separations. The main finding (“D-set Theorem”) is that for any graph, possibly infinite, with maximum degree at most ∆, there is a labelling of minimum span in which all of the labels have the form ∑p i=1 aiki, where the ai’s are integers ≥ 0. We show that the minimum span is a continuous function of the ki’s, and we conjecture that it is piecewise linear with finitely many pieces. Our stronger conjecture is that the coefficients ai can be bounded by a constant depending only on ∆ and p. We offer results in strong support of the conjectures, and we give formulas for the minimum spans of several graphs with general conditions at distance two.