Optimal L(2,1)-labeling of Cartesian products of cycles, with an application to independent domination

The L(2,1)-labeling of a graph is an abstraction of the problem of assigning (integer) frequencies to radio transmitters, such that transmitters that are "close", receive different frequencies, and those that are "very close" receive frequencies that are further apart. The least span of frequencies in such a labeling is referred to as the λ-number of the graph. Let n be odd ≥5, k=(n-3)/2 and let m₀,...,m_k-1, m_k each be a multiple of n. It is shown that λ(Cm₀□···□Cm_k-1) is equal to the theoretical minimum of n-1, where C_r denotes a cycle of length r and "□" denotes the Cartesian product of graphs. The scheme works for a vertex partition of Cm₀□···□Cm_k-1□Cm_k into smallest (independent) dominating sets.