Realization of digraphs in Abelian groups and its consequences

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 $|\Gamma |\ge 4|V(\overrightarrow{G})|$ or if $|V(\overrightarrow{G})|$ is large enough with respect to an arbitrarily fixed $\epsilon >0$ then $|\Gamma |\ge (1+\epsilon )|V(\overrightarrow{G})|$. We show that there exists an injective mapping $\phi$ from $V(\overrightarrow{G})$ to the group $\Gamma$ such that ${\sum }_{x\in V(C)}\phi (x)=0$ 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.

References in corpus (2)

• Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs
• Zero sum partition into sets of the same order and its applications

Cited by in corpus (2)

• Zero-sum partitions of Abelian groups of order $2^n$
• Group irregularity strength of disconnected graphs