Characterization of Completely $k$-Magic Regular Graphs

Abstract

Let $k\in \mathbb{N}$ and $c\in {\mathbb{Z}}_k$, where ${\mathbb{Z}}_1=\mathbb{Z}$. A graph $G=(V(G),E(G))$ is said to be $c$-sum $k$-magic if there is a labeling $\ell :E(G)\to {\mathbb{Z}}_k\setminus \{0\}$ such that ${\sum }_{u\in N(v)}\ell (uv)\equiv c(\mathrm{mod}k)$ for every vertex $v$ of $G$, where $N(v)$ is the neighborhood of $v$ in $G$. We say that $G$ is completely $k$-magic whenever it is $c$-sum $k$-magic for every $c\in {\mathbb{Z}}_k$. In this paper, we characterize all completely $k$-magic regular graphs.

arXiv comment13 pages

References in corpus (1)

• A Conjecture on Zero-sum 3-magic Labeling of 5-regular Graphs

Cited by in corpus (1)

• On Magic Distinct Labellings of Simple Graphs