Irregular labeling on Abelian groups of digraphs

Abstract

Let $\overrightarrow{G}$ be a directed graph of order $n$ with no component of order less than $4$, and let $\Gamma$ be a finite Abelian group such that $|\Gamma |\ge n+6$. We show that there exists a mapping $\psi$ from the arc set $E(\overrightarrow{G})$ of $\overrightarrow{G}$ to an Abelian group $\Gamma$ such that if we define a mapping ${\phi }_{\psi }$ from the vertex set $V(\overrightarrow{G})$ of $\overrightarrow{G}$ to $\Gamma$ by $${\phi }_{\psi }(x)=\sum _{y\in N^{+}(x)}\psi (xy)-\sum _{y\in N^{-}(x)}\psi (yx),(x\in V(\overrightarrow{G})),$$ then ${\phi }_{\psi }$ is injective. Such a labeling $\psi$ is called \textit{irregular}.