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

TL;DR

This paper determines the exact minimum span for a specific edge labeling problem on an infinite octagonal grid, resolving a previously existing gap between known bounds.

Contribution

It proves that the span for the $L(1,2)$-edge labeling of the infinite octagonal grid is exactly 28, closing the gap between earlier bounds.

Findings

Established that $oxed{ ext{span} = 28}$ for the $L(1,2)$-edge labeling of the infinite octagonal grid.

Resolved the open problem by proving the exact value of the labeling span.

Confirmed the previous lower bound and improved the upper bound to match.

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 $f':E(G) \xrightarrow{}\{0,1,\cdots, n\}$ such that $\forall e_1,e_2 \in E(G)$, $\vert f'(e_1)-f'(e_2) \vert \geq h$ when $d'(e_1,e_2)=1$ and $\vert f'(e_1)-f'(e_2) \vert \geq k$ when $d'(e_1,e_2)=2$ where $d'(e_1,e_2)$ denotes the distance between $e_1$ and $e_2$ in $G$. Here $d'(e_1,e_2)=k'$ if there are at least $(k'-1)$ 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 $\lambda'_{h,k}(G)$. 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 $25 \leq \lambda'_{1,2}(T_8) \leq 28$ [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 $\lambda'_{1,2}(T_8)= 28$.