$L(3,2,1)$-LABELING FOR THE PRODUCT OF A COMPLETE GRAPH AND A CYCLE

Abstract

Given a graph $G=(V,E)$, a function $f$ on $V$ is an $L(3,2,1)$-labeling if for each pair of vertices $u,v$ of $G$, it holds that $|f(u)-f(v)|\ge 4-{\text {dist}}(u,v)$. $L(3,2,1)$-labeling number for $G$, denoted by $\lambda_{3,2,1}(G)$, is the minimum span of all $L(3,2,1)$-labeling $f$ for $G$. In this paper, when $G=K_m\Box C_n$ is the Cartesian product of the complete graph $K_m$ and the cycle $C_n$, we show that the lower bound of $\lambda_{3,2,1}(G)$ is $5m-1$ for $m\ge 3$, and the equality holds if and only if $n$ is a multiple of 5. Moreover, we show that $\lambda_{3,2,1}(K_3\Box C_n)=15$ when $n\ge 28$ and $n\not \equiv 0 \pmod 5$.