### $L(3,2,1)$-labeling for Cylindrical grid: the cartesian product of a path and a cycle

#### Abstract

An $L(3,2,1)$-labeling for the graph $G=(V,E)$ is an assignment $f$ of a label to each vertices of $G$ such that $|f(u)-f(v)| \geq 4-k$ when $\textrm{dist}(u,v)=k\leq3$. The $L(3,2,1)$-labeling number, denoted by $\lambda_{3,2,1}(G)$, for $G$ is the smallest number $N$ such that there is an $L(3,2,1)$-labeling for $G$ with span $N$.

In this paper, we compute the $L(3,2,1)$-labeling number $\lambda_{3,2,1}(G)$ when $G$ is a cylindrical grid, which is the cartesian product $P_{m}\square C_{n}$ of the path and the cycle, when $m\geq 4$ and $ n\geq 138$. Especially when $n$ is a multiple of $4$, or $m=4$ and $n$ is a multiple of $6$, then we have $\lambda_{3,2,1}(G)=11$. Otherwise $\lambda_{3,2,1}(G)=12$.

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DOI: http://dx.doi.org/10.11568/kjm.2017.25.2.279

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