L(2,1)L(2,1)-labelings of Cartesian products of two cycles

An L(2,1)L(2,1)-labeling of a graph is an assignment of nonnegative integers to its vertices so that adjacent vertices get labels at least two apart and vertices at distance two get distinct labels. The λλ-number of a graph G  , denoted by λ(G)λ(G), is the minimum range of labels taken over all of its L(2,1)L(2,1)-labelings. We show that the λλ-number of the Cartesian product of any two cycles is 6, 7 or 8. In addition, we provide complete characterizations for the products of two cycles with λλ-number exactly equal to each one of these values.