# Realization of digraphs in Abelian groups and its consequences

## TL;DR

This paper proves that under certain size conditions, directed graphs can be embedded into finite Abelian groups such that each component sums to zero, with applications to group distance magic labelings.

## Contribution

It establishes the existence of injective mappings from directed graphs to Abelian groups with zero-sum conditions on components, extending graph group labelings.

## Key findings

- Existence of injective mappings with zero-sum components
- Conditions on group size relative to graph order
- Applications to group distance magic labelings

## Abstract

Let $\overrightarrow{G}$ be a directed graph with no component of orderless than~$3$, and let $\Gamma$ be a finite Abelian group such that $|\Gamma|\geq 4|V(\overrightarrow{G})|$ or if $|V(\overrightarrow{G})|$ is large enough with respect to an arbitrarily fixed $\varepsilon>0$ then $|\Gamma|\geq (1+\varepsilon)|V(\overrightarrow{G})|$. We show that there exists an injective mapping $\varphi$ from $V(\overrightarrow{G})$ to the group $\Gamma$ such that $\sum_{x\in V(C)}\varphi(x) = 0$ for every connected component $C$ of $\overrightarrow{G}$, where $0$ is the identity element of $\Gamma$. Moreover we show some applications of this result to group distance magic labelings.