Open XOR-magic odd graphs and closed XOR-magic even graphs
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908727855022080 |
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| author | Cichacz, Sylwia Grochowski, Hubert Zuazua, Rita |
| author_facet | Cichacz, Sylwia Grochowski, Hubert Zuazua, Rita |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Open XOR-magic odd graphs and closed XOR-magic even graphs Cichacz, Sylwia Grochowski, Hubert Zuazua, Rita Combinatorics 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. |
| title | Open XOR-magic odd graphs and closed XOR-magic even graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.19278 |