  A mapping $l : E(G) \rightarrow A$, where $A$ is an abelian group which
written additively, is called a labeling of the graph $G$. For every positive
integer $h \geqslant 2$, a graph $G$ is said to be zero-sum $h$-magic if there
is an edge labeling $l$ from $E(G)$ into $\mathbb{Z}_{h} \backslash \{0\}$ such
that $s(v) = \sum_{uv\in E(G)}l(uv) = 0$ for every vertex $v \in V(G)$. In
2014, Saieed Akbari, Farhad Rahmati and Sanaz Zare conjectured that every
5-regular graph admits a zero-sum $3$-magic labeling. In this paper, we
obtained that every 5-regular graph with every edge contains in a triangle must
have a perfect matching, and admits a zero-sum 3-magic labeling, which
partially confirms this conjecture.
