$L(j,k)$-labeling number of Cartesian product of path and cycle

Abstract

For positive numbers $j$ and $k$, an $L(j,k)$-labeling $f$ of $G$ is an assignment of numbers to vertices of $G$ such that $|f(u)-f(v)|\ge j$ if $d(u,v)=1$, and $|f(u)-f(v)|\ge k$ if $d(u,v)=2$. The span of $f$ is the difference between the maximum and the minimum numbers assigned by $f$. The $L(j,k)$-labeling number of $G$, denoted by $\lambda _{j,k}(G)$, is the minimum span over all $L(j,k)$-labelings of $G$. In this article, we completely determine the $L(j,k)$-labeling number ($2j\le k$) of the Cartesian product of path and cycle.