Optimal L(2,1)-labeling of Cartesian products of cycles, with an application to independent domination - Jha, P. K. (2000)
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 lambda -number of the graph, Let n be odd greater than or equal to5, k = (n-3)/2 and let m(o,...,) m(k-1), m(k) each be a multiple of n. It is shown that lambda (C(m0)square...squareC(mk-1)) is equal to the theoretical minimum df n - 1, where C-r denotes a cycle of length r and "square" denotes the Cartesian product of graphs. The scheme works for a vertex partition of C-m0 square...squareC(mk-1) squareC(k) into smallest (independent) dominating sets.