# Improved Lower Bound for L(1, 2)-Edge-Labeling of Infinite 8-Regular Grid

## Document Type

Conference Article

## Publication Title

Lecture Notes in Networks and Systems

## Abstract

Let us take two given integers h and k where h, k≥ 0 and a graph G= (V(G), E(G) ). An L(h, k)-edge-labeling for G is defined as a function f′: E(G) → { 0, 1, ⋯, n} such that ∀ e1, e2∈ E(G), | f′(e1) - f′(e2) | ≥ h when d′(e1, e2) = 1 and | f′(e1) - f′(e2) | ≥ k when d′(e1, e2) = 2 where d′(e1, e2) denotes the distance between e1 and e2 in G. If there is no other way to connect e1 and e2 with less than k′- 1 number of edges in G, then d′(e1, e2) = k′. The objective is to find the span which is the least value of n obtained over all such L(h, k)-edge-labeling for G, and it is denoted as λh,k′(G). Motivated from the channel assignment problem in wireless network, L(h, k)-edge-labeling has been studied for various infinite regular grid graphs. For infinite 8-regular grid T8, it is known in the existing literature that 25≤λ1,2′(T8)≤28. But the lower and upper bounds obtained there are not identical. Here, we improve the lower bound and prove that λ1,2′(T8)≥28 and eventually it is proved that λ1,2′(T8)=28.

## First Page

261

## Last Page

273

## DOI

10.1007/978-981-99-3080-7_20

## Publication Date

1-1-2023

## Recommended Citation

Koley, Subhasis and Ghosh, Sasthi C., "Improved Lower Bound for L(1, 2)-Edge-Labeling of Infinite 8-Regular Grid" (2023). *Conference Articles*. 571.

https://digitalcommons.isical.ac.in/conf-articles/571