Characterization of Completely $k$-Magic Regular Graphs

TL;DR

This paper characterizes all regular graphs that are completely $k$-magic, meaning they admit a specific edge labeling for every possible sum residue modulo $k$, extending the understanding of graph labelings.

Contribution

It provides a complete characterization of regular graphs that are completely $k$-magic, a new class of graphs with uniform sum-labeling properties.

Findings

Identifies conditions under which regular graphs are completely $k$-magic.

Classifies all such regular graphs for given $k$.

Extends the theory of graph labelings and magic 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 $\ell:E(G) \rightarrow \mathbb{Z}_k \setminus \{0\}$ such that $\sum_{u \in N(v)} \ell(uv) \equiv c \pmod{k}$ 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.