Polyominoes with maximal number of deep holes
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908757214101504 |
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| author | Baralic, Djordje Uppal, Shiven |
| author_facet | Baralic, Djordje Uppal, Shiven |
| contents | In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any other hole and from the outer boundary of the $n$-tile. Using the versatile application of Pick's theorem, we establish the lower and the upper bound for $h_n$, and show that $h_n=\frac{n}{3}+o(n)$ asymptotically. To further develop these results, we compute $h_n$ as a function of $n$ for an infinite subset of positive integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06840 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polyominoes with maximal number of deep holes Baralic, Djordje Uppal, Shiven Combinatorics Optimization and Control 05B50 In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any other hole and from the outer boundary of the $n$-tile. Using the versatile application of Pick's theorem, we establish the lower and the upper bound for $h_n$, and show that $h_n=\frac{n}{3}+o(n)$ asymptotically. To further develop these results, we compute $h_n$ as a function of $n$ for an infinite subset of positive integers. |
| title | Polyominoes with maximal number of deep holes |
| topic | Combinatorics Optimization and Control 05B50 |
| url | https://arxiv.org/abs/2601.06840 |