Labeling Products of Complete Graphs with a Condition at Distance Two
John P. Georges, David W. Mauro, Melanie Stein · SIAM Journal on Discrete Mathematics · 2001
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.