L(j,k)- and circular L(j,k)-labellings for the products of complete graphs

Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λj,k-number of G, denoted by λj,k(G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)|m≥j if u and v are adjacent; and |f(u)−f(v)|m≥k if u and v are at distance two apart, where |x|m=min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σj,k-number of G and denoted by σj,k(G). The λj,k-numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λj,k-numbers of direct products of two complete graphs and the σj,k-numbers of direct products and Cartesian products of two complete graphs.