L(3,2,1)-labeling of certain planar graphs

Given a graph G = (V, E) of maximum degree Delta, denoting by d(x, y) the distance in G between nodes x, y is an element of V, an L(3, 2, 1)-labeling of G is an assignment l from V to the set of non-negative integers such that |l(x)-l(y)| >= 3 if x and y are adjacent, |l(x)-l(y)| >= 2 if d(x, y) = 2, and |l(x)-l(y)| >= 1 if d(x, y) = 3, for all x and y in V. The L(3, 2, 1)-number lambda(G) is the smallest positive integer such that G admits an L(3, 2, 1)-labeling with labels from {0, 1, & mldr;, lambda(G)}. In this paper, the L(3, 2, 1)-number of certain planar graphs is determined, proving that it is linear in Delta, although the general upper bound for the L(3, 2, 1)-number of planar graphs is quadratic in Delta.