On the $\lambda$-Number of $Q_n $ and Related Graphs

An $L(2, l)$-labeling of graph $G$ is an integer labeling of $V(G)$ such that adjacent vertices have labels that differ by at least 2 and such that vertices distance 2 apart have labels that differ by at least 1. The $\lambda$-number of $G, \lambda(G)$, is the minimum range over all $L(2, 1)$-labelings. We examine the properties of $\lambda$-labelings of the $n$-cube $Q_n$. Griggs and Yeh have determined $\lambda(Q_n)$ for $n \leq 5$ and have established $n + 3 \leq \lambda(Q_n) \leq 2n + 1$ for $n \geq 6$. We modify a technique used in coding theory to improve the upper bound. We also examine the $\lambda$-labelings of related graphs, such as the subdivision of the $n$-cube and the Cartesian products of paths.