Optimal L(2, 1)-labeling of strong products of cycles [transmitter frequency assignment]

The L(2, 1)-labeling of a graph is an abstraction of assigning integer frequencies to radio transmitters such that i) transmitters that are one unit of distance apart receive frequencies that differ by at least two, and ii) transmitters that are two units of distance apart receive frequencies that differ by at least one. The least span of frequencies in such a labeling is referred to as the λ-number of the graph. It is shown that if k⩾1 and m₀, ..., m_k-1 are each a multiple of 3^k+2, then λ(Cm₀ □...□Cm_k-1) is equal to the theoretical minimum of 3^k+1, where C_i denotes the cycle of length i and □ denotes the strong product of graphs