The $L(3,2,1)$Labeling Problem for Trees 
Received:July 29, 2019 Revised:March 17, 2020 
Key Words:
channel assignment $L(3,2,1)$labeling trees diameter

Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11601265) and the HighLevel Talent Innovation and Entrepreneurship Project of Quanzhou City (Grant No.2017Z033). 

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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 nonnegative 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 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. In this article, we characterize the $L(3, 2, 1)$labeling numbers of trees with diameter at most 6. 
Citation: 
DOI:10.3770/j.issn:20952651.2020.05.003 
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