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 $x,y\in V$, an $L(3,2,1)$-labeling of G is an assignment l from V to the set of non-negative integers such that $\left|l\right(x)-l(y\left)\right|\ge 3$ if x and y are adjacent, $\left|l\right(x)-l(y\left)\right|\ge 2$ if $d(x,y)=2$, and $\left|l\right(x)-l(y\left)\right|\ge 1$ if $d(x,y)=3$, for all x and y in V. The $L(3,2,1)$-number $\lambda \left(G\right)$ is the smallest positive integer such that G admits an $L(3,2,1)$-labeling with labels from $\{0,1,\dots ,\lambda (G\left)\right\}$.

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.