Xueqin Zhang

In this paper, we extend the result for the case where every vertex needs one label to the case where every vertex needs $k$ labels and $k$ is a positive integer number. We also give the definition of the composite graph $G[K_{k}]$ and the definition of $L$(2, 1)-labeling of the composite graph $G[K_{k}]$, where $K_{k}$ is a complete graph of $k$ vertices. We determine the $L$(2, 1)-labeling number of $P_{n}[K_{k}]$, where $P_{n}$ is a path of $n$ vertices. Then we show that $\lambda(T[K_{k}])$ is $(\Delta+2)k-1$ or $(\Delta+2)k$ for any tree $T$ of maximum degree $\Delta$, and present a polynomial time algorithm to determine $\lambda(T[K_{k}])$. Finally, we obtain partial results for the $\lambda$-numbers of $C_{n}[K_{k}]$, where $C_{n}$ is a cycle of $n$ vertices.