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

An L(2,1)-labeling of a graph G is an assignment of labels from {0,1,…,λ} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. The λ-number λ(G) of G is the minimum value λ such that G admits an L(2,1)-labeling. Let G×H denote the direct product of G and H. We compute the λ-numbers for each of C7i×C7j, C11i×C11j×C11k, P4×Cm, and P5×Cm. We also show that for n⩾6 and m⩾7, λ(Pn×Cm)=6 if and only if m=7k, k⩾1. The results are partially obtained by a computer search.