L(2,1)Circular Labelings of Cartesian Products of Complete Graphs 
Received:December 17, 2006 Revised:September 04, 2007 
Key Word:
$\lambda_{2,1}$number $\sigma_{2,1}$number Cartesian product.

Fund ProjectL:the National Natural Science Foundation of China (No.10671033); the Science Foundation of Southeast University (No.XJ0607230); the Natural Science Foundation of Nantong University (No.08Z003). 

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Abstract: 
For positive integers $j$ and $k$ with $j\geq k$, an $L(j,k)$labeling of a graph $G$ is an assignment of nonnegative integers to $V(G)$ such that the difference between labels of adjacent vertices is at least $j$, and the difference between labels of vertices that are distance two apart is at least $k$. The span of an $L(j,k)$labeling of a graph $G$ is the difference between the maximum and minimum integers it uses. The $\lambda_{j,k}$number of $G$ is the minimum span taken over all $L(j,k)$labelings of $G$. An $m$$(j,k)$circular labeling of a graph $G$ is a function $f: V(G)\rightarrow \{0,1,2,\ldots,m1\}$ such that $f(u)f(v)_{m}\geq j$ if $u$ and $v$ are adjacent; and $f(u)f(v)_{m}\geq k$ if $u$ and $v$ are at distance two, where $x_{m}=\min\{x,mx\}$. The minimum integer $m$ such that there exists an $m$$(j,k)$circular labeling of $G$ is called the $\sigma_{j,k}$number of $G$ and is denoted by $\sigma_{j,k}(G)$. This paper determines the $\sigma_{2,1}$number of the Cartesian product of any three complete graphs. 
Citation: 
DOI:10.3770/j.issn:1000341X.2009.01.012 
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