The $L(3,2,1)$-Labeling Problem for Trees

Abstract:

      An $L(3,2,1)$-labeling of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all non-negative integers (labels) such that $|f(u)-f(v)| \geq 3$ if $d(u, v) = 1$, $|f(u) - f(v)| \geq 2$ if $d(u, v) = 2$ and $|f(u)-f(v)| \geq 1$ if $d(u, v) = 3$. For a non-negative 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. In this article, we characterize the $L(3, 2, 1)$-labeling numbers of trees with diameter at most 6.