A new approach to the L(2,1)-labeling of some products of graphs

The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex labelings of graphs. An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|≥ 2 if d(x,y) =1 and |f(x)-f(y)| ≥ 1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ (G) of G is the smallest number k such that G has an L(2,1)-labeling with max｛f(v) : v Є V (G)｝= k. In this paper, we develop a dramatically new approach on the analysis of the adjacency matrices of the graphs to estimate the upper bounds of λ-numbers of the four standard graph products. By the new approach, we can achieve more accurate results and with significant improvement of the previous bounds. Copyright © 2008 IEEE Circuits and Systems Society.