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

Abstract

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.