Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sylwia Cichacz, Hubert Grochowski, Rita Zuazua

TL;DR

The paper resolves an open problem by proving that for every n>3 there exist an odd-regular open XOR-magic graph and an even-regular closed XOR-magic graph on 2^n vertices, using MILP-based search for base graphs and algebraic techniques to guide constructions. A key methodological contribution is the det(A(G)) ≡ 0 (mod 2) criterion for open XOR-magic labelings and the Smith normal form analysis of adjacency matrices for circulant graphs, which clarifies nonexistence in several classes. The main constructive theorem is established by induction on n via Cartesian and other graph products, leveraging explicit base graphs found with optimized MILP encodings. The work also connects open and closed variants through graph complements and demonstrates spectral viewpoints for open XOR-magic labeling.

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.

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

TL;DR

The paper resolves an open problem by proving that for every n>3 there exist an odd-regular open XOR-magic graph and an even-regular closed XOR-magic graph on 2^n vertices, using MILP-based search for base graphs and algebraic techniques to guide constructions. A key methodological contribution is the det(A(G)) ≡ 0 (mod 2) criterion for open XOR-magic labelings and the Smith normal form analysis of adjacency matrices for circulant graphs, which clarifies nonexistence in several classes. The main constructive theorem is established by induction on n via Cartesian and other graph products, leveraging explicit base graphs found with optimized MILP encodings. The work also connects open and closed variants through graph complements and demonstrates spectral viewpoints for open XOR-magic labeling.

Abstract

XOR-magic graph labelings form a special subclass of group distance magic labelings. A simple connected graph of order is called an open (respectively, closed) XOR-magic graph of power if its vertices can be labeled bijectively with vectors from such that the sum (over ) 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 , there exists an odd-regular open XOR-magic graph of power and an even-regular closed XOR-magic graph of power . We also show some applications of the spectra of graphs for an open XOR-magic labeling.
Paper Structure (4 sections, 21 theorems, 27 equations, 5 figures)

This paper contains 4 sections, 21 theorems, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Algebraic approach and its application to Cayley graphs
  3. Odd regular open XOR-magic graphs and even regular closed XOR-magic graphs
  4. Conclusions

Key Result

Theorem 2

Let $G$ be an $r$-regular graph on $n$ vertices, where $r$ is odd. If $\Gamma$ is an Abelian group of order $n$ with $|I(\Gamma)|=1$, then $G$ is not $\Gamma$-distance magic

Figures (5)

  • Figure 1: A $(\mathop\mathbb{Z}\nolimits_2)^2$-distance magic labeling of $K_{2,2}$ and a $(\mathop\mathbb{Z}\nolimits_2)^3$-distance magic labeling of $K_{4,4}$
  • Figure 2: First MILP model for finding odd regular open XOR-magic graphs (parameters: $n \in \mathbb{N}_+, V = [1,2^n], d \in \mathbb{N}_+$ - odd number, $\ell: V \to (\mathbb{Z}_2)^n$ - fixed bijection)
  • Figure 3: Open XOR-Magic $d$-regular graphs of order $2^4$ for $d \in \{5,7,9,11\}$.
  • Figure 4: Second MILP model for finding odd regular open XOR-magic graphs (parameters: $n \in \mathbb{N}_+, t \in \mathbb{N}_+, t \geq 1, t \leq n, V = [1,2^n], d \in \mathbb{N}_+$ - odd number, $\ell: V \to (\mathbb{Z}_2)^n$ - fixed bijection, $S_t = (d^t, d^{t-1}, \ldots, d), S_{end} = ( d^{n - \left(\lceil \frac{n}{t} \rceil - 1 \right)t}, \ldots, d^2, d)$)
  • Figure 5: A $4$-regular closed XOR-magic graph of order $16$

Theorems & Definitions (24)

  • Theorem 2: CicFro
  • Theorem 5
  • Theorem 7
  • Lemma 8
  • Corollary 9
  • Theorem 10
  • Corollary 11
  • Corollary 12
  • Corollary 13
  • Corollary 14
  • ...and 14 more