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.