$L(j,k)$-labelings of Cartesian Products of Three Complete Graphs

Abstract

Given positive integers $j$ and $k$ with $j\ge k$, an {L$(j,k)$-labeling} of a graph $G$ assigns nonnegative integers to $V(G)$ such that adjacent vertices’ labels differ by at least $j$, and vertices distance two apart have labels differing by at least $k$. The span of an L$(j,k)$-labeling is the difference between the maximum and minimum assigned integers. The ${\lambda }_{j,k}$-number of $G$ is the minimum span over all L$(j,k)$-labelings of $G$. This paper investigates the ${\lambda }_{j,k}$-numbers of Cartesian products of three complete graphs.