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

Given a graph 𝐺 = (𝑉 , 𝐸 ) of maximum degree 𝛥, denoting by 𝑑 (𝑥, 𝑦) the distance in 𝐺 between nodes 𝑥, 𝑦 ∈ 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 𝜆(𝐺) is the smallest positive integer such that 𝐺 admits an 𝐿(3, 2, 1)-labeling with labels from {0, 1, ... , 𝜆(𝐺)}. In this paper, the 𝐿(3, 2, 1)-number of certain planar graphs is determined, proving that it is linear in 𝛥, although the general upper bound for the 𝐿(3,2,1)-number of planar graphs is quadratic in 𝛥.