Pertfect matching and zero-sum 3-magic labeling

TL;DR

This paper investigates zero-sum 3-magic labelings of 5-regular graphs, proving that those containing triangles have perfect matchings and admit such labelings, thus providing partial support for a longstanding conjecture.

Contribution

It establishes that 5-regular graphs with triangles have perfect matchings and can be labeled zero-sum 3-magic, partially confirming the conjecture for this class.

Findings

Graphs with triangles have perfect matchings.

Such graphs admit zero-sum 3-magic labelings.

Partial confirmation of the 5-regular graph conjecture.

Abstract

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.