L(2,1)-labeling of the strong product of paths and cycles

An L(2,1)-labeling of a graph G = (V, E) is a function f from the vertex set V(G) to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two and the labels on vertices at distance two differ by at least one. The span of f is the difference between the largest and the smallest numbers in f(V). The λ-number of G, denoted by λ(G), is the minimum span over all L(2,1)-labelings of G. We consider the λ-number of Pn⊠C m and for n ≤ 11 the λ-number of Cn⊠Cm. We determine λ-numbers of graphs of interest with the exception of a finite number of graphs and we improve the bounds on the λ-number of Cn⊠Cm, m ≥ 24 and n ≥ 26.