The L(2, 1)-Labeling Problem on Oriented Regular Grids

TL;DR

This paper investigates the L(2, 1)-labeling problem on specific oriented regular grids, providing exact values of the minimum maximum label for squared, triangular, and hexagonal grids.

Contribution

It computes the exact L(2, 1)-labeling numbers for squared, triangular, and hexagonal grids, advancing understanding of labelings on these structures.

Findings

Exact values of λ(G) for squared grids

Exact values of λ(G) for triangular grids

Exact values of λ(G) for hexagonal grids

Abstract

The L(2, 1)-labeling of a digraph G is a function f from the node set of $G$ to the set of all nonnegative integers such that $|f(x)-f(y)| \geq 2$ if $x$ and $y$ are at distance 1, and $f(x)=f(y)$ if $x$ and $y$ are at distance 2, where the distance from vertex $x$ to vertex $y$ is the length of a shortest dipath from $x$ to $y$. The minimum of the maximum used label over all $L(2, 1)$-labelings of $G$ is called $\lambda(G)$. In this paper we study the L(2, 1)-labeling problem on squared, triangular and hexagonal grids and for them we compute the exact values of $\lambda$.