Realization of digraphs in Abelian groups and its consequences

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

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