Open XOR-magic odd graphs and closed XOR-magic even graphs

TL;DR

This paper proves the existence of certain XOR-magic graphs with specific regularities for all powers greater than three, using computational methods and spectral graph theory insights.

Contribution

It establishes the existence of odd-regular open XOR-magic graphs and even-regular closed XOR-magic graphs for all powers greater than three, answering a previously open question.

Findings

Existence of odd-regular open XOR-magic graphs for all n>3

Existence of even-regular closed XOR-magic graphs for all n>3

Application of spectral graph theory to XOR-magic labelings

Abstract

XOR-magic graph labelings form a special subclass of group distance magic labelings. A simple connected graph of order $2^n$ is called an open (respectively, closed) XOR-magic graph of power $n$ if its vertices can be labeled bijectively with vectors from $(\mathbb{Z}_2)^n$ such that the sum (over $(\mathbb{Z}_2)^n$) of labels in each open (respectively, closed) neighborhood of every vertex is equal to the zero vector. In one paper, Batal asked whether there exists any odd-regular open XOR-magic graph or any even-regular closed XOR-magic graph. In this paper, with partial help of MILP solver, we answer this question in the affirmative. More precisely, we prove that for every integer $n>3$, there exists an odd-regular open XOR-magic graph of power $n$ and an even-regular closed XOR-magic graph of power $n$. We also show some applications of the spectra of graphs for an open XOR-magic labeling.