NEW $L(j,k)$-LABELINGS FOR DIRECT PRODUCTS OF COMPLETE GRAPHS

Abstract

An $L(j,k)$-$labeling$ of a graph is a vertex labeling such that the difference between the labels of adjacent vertices is at least $j$ and that between vertices separated by a distance 2 is at least $k$. The minimum of the spans of all $L(j,k)$-labelings of $G$ is denoted by $\lambda_k^j(G)$. Recently, Haque and Jha [16] proved that if $G$ is a multiple direct product of complete graphs, then $\lambda_k^j(G)$ coincides with the trivial lower bound $(N-1)k$, where $N$ is the order of $G$ and $\frac{j}{k}$ is within a certain bound. In this paper, we suggest a new labeling method for such a graph $G$. With this method, we extend the range of $\frac{j}{k}$ such that $\lambda_k^j(G)= (N-1)k$ holds. Moreover, we obtain the upper bound of $\lambda_k^j(G)$ for the remaining cases in the range $\frac{j}{k}$.