Taiwanese Journal of Mathematics


Ma-Lian Chia, David Kuo, Hong-ya Liao, Cian-Hui Yang, and Roger K. Yeh

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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.

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Taiwanese J. Math., Volume 15, Number 6 (2011), 2439-2457.

First available in Project Euclid: 18 July 2017

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Primary: 05

$L(3,2,1)$-labeling tree Cartesian product power path cycle


Chia, Ma-Lian; Kuo, David; Liao, Hong-ya; Yang, Cian-Hui; Yeh, Roger K. $L(3,2,1)$-LABELING OF GRAPHS. Taiwanese J. Math. 15 (2011), no. 6, 2439--2457. doi:10.11650/twjm/1500406480. https://projecteuclid.org/euclid.twjm/1500406480

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