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

DOI：10.3770/j.issn:2095-2651.2020.05.003

 作者 单位 张小玲 泉州师范学院数学与计算机科学学院, 福建 泉州 362000

图$G$的一个$L(3,2,1)$-标号是指从$V(G)$到非负整数集上的一个函数$f$,满足:当$d(u,v)=1$时, $|f(u)-f(v)|\ge 3$,当$d(u,v)=2$时, $|f(u)-f(v)|\ge 2$,当$d(u,v)=3$时, $|f(u)-f(v)|\ge 1$. 若一个$L(3,2,1)$-标号中的所有像元素都不超过整数$k$,则称之为图$G$的$k$-$L(3,2,1)$-标号. 图$G$的$L(3,2,1)$-标号数, 记作$\lambda 3,2,1(G)$,是使得图$G$存在$L(3,2,1)$-标号的最小整数$k$. 本文完全刻画了直径不超过6的树的$L(3,2,1)$-标号数.

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.