Labeling Products of Complete Graphs with a Condition at Distance Two

For integers $j \geq k$, an L(j,k)-labeling of a graph G is an integer labeling of the vertices in V(G) such that adjacent vertices receive integers which differ by at least j, and vertices which are distance two apart receive labels which differ by at least k. We determine $\lambda^j_k(K_n \times K_m)$ for all j,k,m,n, and $\lambda^{2}_1(K^q_{p^r})$ for $3 \leq q < p$, p prime.