Distance two labeling of direct product of paths and cycles

Abstract

Suppose that $[n]=\left\{0,1,2,...,n\right\}$ is a set of non-negative integers and $h,k \in [n]$. The $L(h,k)$-labeling of graph $G$ is the function $l:V(G)\rightarrow[n]$ such that $\left|l(u)-l(v)\right|\geq h$ if the distance $d(u,v)$ between $u$ and $v$ is one and $\left|l(u)-l(v)\right| \geq k$ if the distance $d(u,v)$ is two. Let $L(V(G))=\left\{l(v): v \in V(G)\right\}$ and let $p$ be the maximum value of $L(V(G)).$ Then $p$ is called $\lambda_h^k-$number of $G$ if $p$ is the least possible member of $[n]$ such that $G$ maintains an $L(h,k)-$labeling. In this paper, we establish $\lambda_1^1-$ numbers of $P _m \times C_n$ graphs for all $m \geq 2$ and $n\geq 3$.