No-hole $\lambda$-$L(k, k-1, \ldots, 2, 1)$-labeling for Square Grid

TL;DR

This paper establishes a lower bound and proposes a near-optimal no-hole labeling scheme for the $ ext{L}(k, k-1, ..., 2, 1)$-labeling problem on square grids, achieving an approximation ratio of at most 9/8.

Contribution

It introduces a new lower bound and a formula for a no-hole labeling of square grids with a guaranteed approximation ratio, advancing graph labeling theory.

Findings

Derived a lower bound for the labeling number on square grids.

Proposed a labeling formula with approximation ratio ≤ 9/8.

Provided a no-hole labeling scheme that uses all labels at least once.

Abstract

Given a fixed $k$ $\in$ $\mathbb{Z}^+$ and $\lambda$ $\in$ $\mathbb{Z}^+$, the objective of a $\lambda$-$L(k, k-1, \ldots, 2, 1)$-labeling of a graph $G$ is to assign non-negative integers (known as labels) from the set $\{0, \ldots, \lambda-1\}$ to the vertices of $G$ such that the adjacent vertices receive values which differ by at least $k$, vertices connected by a path of length two receive values which differ by at least $k-1$, and so on. The vertices which are at least $k+1$ distance apart can receive the same label. The smallest $\lambda$ for which there exists a $\lambda$-$L(k, k-1, \ldots, 2, 1)$-labeling of $G$ is known as the $L(k, k-1, \ldots, 2, 1)$-labeling number of $G$ and is denoted by $\lambda_k(G)$. The ratio between the upper bound and the lower bound of a $\lambda$-$L(k, k-1, \ldots, 2, 1)$-labeling is known as the approximation ratio. In this paper a lower bound on the value of the labeling number for square grid is computed and a formula is proposed which yields a $\lambda$-$L(k, k-1, \ldots, 2, 1)$-labeling of square grid, with approximation ratio at most $\frac{9}{8}$. The labeling presented is a no-hole one, i.e., it uses each label from $0$ to $\lambda-1$ at least once.