The L(2, 1)-labeling on planar graphs

Given non-negative integers j and k, an L( j, k)-labeling of a graph G is a function f from the vertex set V (G) to the set of all non-negative integers such that | f (x) − f (y)| ≥ j if d(x, y) = 1 and | f (x) − f (y)| ≥ k if d(x, y) = 2. The L( j, k)-labeling number λ j,k is the smallest number m such that there is an L( j, k)-labeling with the largest value m and the smallest label 0. This paper presents upper bounds on λ2,1 and λ2,1 of a graph G in terms of the maximum degree of G for several classes of planar graphs. These bounds are the same as or better than previous results for the maximum degree less than or equal to 4. c © 2006 Elsevier Ltd. All rights reserved.