$L(3,2,1)$-LABELING OF GRAPHS

Abstract

Given a graph $G,$ an $L(3,2,1)$-labeling of $G$ is a function $f$ from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(u)-f(v)|\geqslant 1$ if $d(u,v) = 3,$ $|f(u)-f(v)| \geqslant 2$ if $d(u,v) = 2$ and $|f(u)-f(v)| \geqslant 3$ if $d(u,v) = 1$. For a nonnegative integer $k$, a $k$-$L(3,2,1)$-labeling is an $L(3,2,1)$-labeling such that no label is greater than $k$. The $L(3,2,1)$-labeling number of $G$, denoted by $\lambda_{3,2,1}(G)$, is the smallest number $k$ such that $G$ has a $k$-$L(3,2,1)$-labeling. We study the $L(3,2,1)$-labelings of graphs in this paper. We give upper bounds for the $L(3,2,1)$-labeling numbers of general graphs and trees, and consider the $L(3,2,1)$-labeling numbers of several classes of graphs, such as the Cartesian product of paths and cycles, and the powers of paths.