L(h,k)L(h,k)-labelings of Hamming graphs

Given integers c≥0c≥0 and h≥k≥1h≥k≥1, a cc-L(h,k)L(h,k)-labeling of a graph GG is a mapping f:V(G)→{0,1,2,…,c}f:V(G)→{0,1,2,…,c} such that |f(u)−f(v)|≥h|f(u)−f(v)|≥h if dG(u,v)=1dG(u,v)=1 and |f(u)−f(v)|≥k|f(u)−f(v)|≥k if dG(u,v)=2dG(u,v)=2. The L(h,k)L(h,k)-number λh,k(G)λh,k(G) of GG is the minimum cc such that GG has a cc-L(h,k)L(h,k)-labeling. The Hamming graph is the Cartesian product of complete graphs. In this paper, we study L(h,k)L(h,k)-labeling numbers of Hamming graphs. In particular, we determine λh,k(Knq) for 2≤q≤p2≤q≤p with h/k≤n−q+1h/k≤n−q+1 or 2≤q≤p2≤q≤p with h/k≥qn−2q+2h/k≥qn−2q+2 or q=p+1q=p+1 with h/k≤n/ph/k≤n/p, where pp is the minimum prime factor of nn.