Distance-two labelings of graphs

For given positive integers j≥k, an L(j,k)-labeling of a graph G is a function f:V(G)→{0,1,2,...} such that f(u)-f(v) ≥j when d_G(u,v)=1 and f(u)-f(v) ≥k when d_G(u,v)=2. The L(j,k)-labeling number λ_j,k (G) of G is defined as the minimum m such that there is an L(j,k)- labeling f of G with f(V(G))⊆{0,1,2,...,m}. For a graph G of maximum degree Δ≥1 it is the case that λ_j,k (G)≥j+(Δ-1)k. The purpose of this paper is to study the structures of graphs G with maximum degree Δ≥1 and λ_j,k(G)=j+(Δ-1)k.