Optimal $L(1,2)$-edge Labeling of Infinite Octagonal Grid

Abstract

For two given non-negative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G=(V(G),E(G))$ is a function $f^{\prime }:E(G)\to \{0,1,\cdots ,n\}$ such that $\forall e_1,e_2\in E(G)$, $|f^{\prime }(e_1)-f^{\prime }(e_2)|\ge h$ when $d^{\prime }(e_1,e_2)=1$ and $|f^{\prime }(e_1)-f^{\prime }(e_2)|\ge k$ when $d^{\prime }(e_1,e_2)=2$ where $d^{\prime }(e_1,e_2)$ denotes the distance between $e_1$ and $e_2$ in $G$. Here $d^{\prime }(e_1,e_2)=k^{\prime }$ if there are at least $(k^{\prime }-1)$ number of edges in $E(G)$ to connect $e_1$ and $e_2$ in $G$. The objective is to find \textit{span} which is the minimum $n$ over all such $L(h,k)$-edge labeling and is denoted as ${\lambda }_{h,k}^{\prime }(G)$. Motivated by the channel assignment problem in wireless cellular network, $L(h,k)$-edge labeling problem has been studied in various infinite regular grids. For infinite regular octagonal grid $T_8$, it was proved that $25\le {\lambda }_{1,2}^{\prime }(T_8)\le 28$ [Tiziana Calamoneri, International Journal of Foundations of Computer Science, Vol. 26, No. 04, 2015] with a gap between lower and upper bounds. In this paper we fill the gap and prove that ${\lambda }_{1,2}^{\prime }(T_8)=28$.