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 dG(u,v)=1 and |f(u)−f(v)|≥k when dG(u,v)=2. The L(j,k)-labeling number λj,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 λj,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 λj,k(G)=j+(Δ−1)k.