Improved Bounds on the Span of $L(1,2)$-edge Labeling of Some Infinite Regular Grids

TL;DR

This paper determines exact bounds for the span of $L(1,2)$-edge labeling on infinite regular grids, resolving longstanding open questions for hexagonal and square grids and improving bounds for triangular and octagonal grids.

Contribution

It provides exact values for the span on hexagonal and square grids and tighter bounds for triangular and octagonal grids, advancing understanding of edge labelings on infinite grids.

Findings

$oxed{ ext{For hexagonal grid } T_3, ext{ span } ext{is } 7}$

$oxed{ ext{For square grid } T_4, ext{ span } ext{is } 11}$

$oxed{ ext{Lower bounds for } T_6 ext{ and } T_8 ext{ improved to } 18 ext{ and } 26$

Abstract

For two given nonnegative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G$ is the assignment of labels $\{0,1, \cdots, n\}$ to the edges so that two edges having a common vertex are labeled with difference at least $h$ and two edges not having any common vertex but having a common edge connecting them are labeled with difference at least $k$. The span $\lambda'_{h,k}{(G)}$ is the minimum $n$ such that $G$ admits an $L(h,k)$-edge labeling. Here our main focus is on finding $\lambda'_{1,2}{(G)}$ for $L(1,2)$-edge labeling of infinite regular hexagonal ($T_3$), square ($T_4$), triangular ($T_6$) and octagonal ($T_8$) grids. It was known that $7 \leq \lambda'_{1,2}{(T_3)} \leq 8$, $10 \leq \lambda'_{1,2}{(T_4)} \leq 11$, $16 \leq \lambda'_{1,2}{(T_6)} \leq 20$ and $25 \leq \lambda'_{1,2}{(T_8)} \leq 28$. Here we settle two long standing open questions i.e. $\lambda'_{1,2}{(T_3)}$ and $\lambda'_{1,2}{(T_4)}$. We show $\lambda'_{1,2}{(T_3)} =7$, $\lambda'_{1,2}{(T_4)}= 11$. We also improve the bound for $T_6$ and $T_8$ and prove $\lambda'_{1,2}{(T_6)} \geq 18$, $ \lambda'_{1,2}{(T_8)} \geq 26$.