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 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 ∆ ≥ 1 it is the case that λ j,k(G) ≥ j + (∆− 1)k. The purpose of this paper is to study the structures of graphs G with maximum degree ∆ ≥ 1 and λ j,k(G) = j + (∆ − 1)k. c © 2003 Elsevier Science Ltd. All rights reserved.