An infinite family of counterexamples to a conjecture on distance magic labeling
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arXiv
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| Format: | Preprint |
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2024
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| author | Ebrahem, Ehab Hoory, Shlomo Kotlar, Dani |
| author_facet | Ebrahem, Ehab Hoory, Shlomo Kotlar, Dani |
| contents | This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study the problem of characterizing the cases where it is possible to find a partition of the set $\{1,2,\ldots,n\}$ into $k$ subsets of respective sizes $p_1,\dots,p_k$, such that the element sum in each subset is equal. Using a computerized search we found examples showing that the necessary condition, $\sum_{i=1}^{p_1+\cdots+p_j} (n-i+1)\ge j{\binom{n+1}{2}}/k$ for all $j=1,\ldots,k$, is not generally sufficient, refuting a past conjecture. Moreover, we show that there are infinitely many such counter-examples. The question whether there is a simple characterization is left open and for all we know the corresponding decision problem might be NP-complete. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00807 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An infinite family of counterexamples to a conjecture on distance magic labeling Ebrahem, Ehab Hoory, Shlomo Kotlar, Dani Combinatorics This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study the problem of characterizing the cases where it is possible to find a partition of the set $\{1,2,\ldots,n\}$ into $k$ subsets of respective sizes $p_1,\dots,p_k$, such that the element sum in each subset is equal. Using a computerized search we found examples showing that the necessary condition, $\sum_{i=1}^{p_1+\cdots+p_j} (n-i+1)\ge j{\binom{n+1}{2}}/k$ for all $j=1,\ldots,k$, is not generally sufficient, refuting a past conjecture. Moreover, we show that there are infinitely many such counter-examples. The question whether there is a simple characterization is left open and for all we know the corresponding decision problem might be NP-complete. |
| title | An infinite family of counterexamples to a conjecture on distance magic labeling |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.00807 |