On Magic Distinct Labellings of Simple Graphs

TL;DR

This paper explores the complete structure and enumeration of all magic labelings of simple graphs, focusing on the complex case of distinct labelings, with detailed combinatorial proofs and applications to regular graphs.

Contribution

It provides a comprehensive construction and enumeration method for all magic labelings, especially distinct ones, of simple graphs, including detailed combinatorial proofs and applications to regular graphs.

Findings

Complete construction of all magic labelings of a graph

Enumeration of magic distinct labelings

Application to three regular graphs

Abstract

A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave remarkable results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph $G$. The idea is illustrated in detail by dealing with three regular graphs. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct labellings.