Distance two labelling and direct products of graphs

An L(2,1)L(2,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G   such that vertices at distance two get different numbers and adjacent vertices get numbers which are at least two apart. The L(2,1)L(2,1)-labelling number of a graph G  , λ(G)λ(G), is the minimum range of labels over all L(2,1)L(2,1)-labellings of G. Given two graphs G and H, the direct product of G and H   is the graph G×HG×H with vertex set V(G)×V(H)V(G)×V(H) in which two vertices (x,y)(x,y) and (x′,y′)(x′,y′) are adjacent if and only if xx′∈E(G)xx′∈E(G) and yy′∈E(H)yy′∈E(H). In this paper, we completely determine the L(2,1)L(2,1)-labelling numbers of Km×KnKm×Kn for m,n⩾2m,n⩾2, and Km×PnKm×Pn for m⩾3m⩾3, n⩾1n⩾1, where PnPn is the path of length n  . The L(2,1)L(2,1)-labelling numbers of Km×CnKm×Cn for m⩾3m⩾3 and some special values of n   are also determined, where CnCn is the cycle of length n.