Domination by kings is oddly even
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929439094341632 |
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| author | Moore, Cristopher Mertens, Stephan |
| author_facet | Moore, Cristopher Mertens, Stephan |
| contents | The $m \times n$ king graph consists of all locations on an $m \times n$ chessboard, where edges are legal moves of a chess king. %where each vertex represents a square on a chessboard and each edge is a legal move. Let $P_{m \times n}(z)$ denote its domination polynomial, i.e., $\sum_{S \subseteq V} z^{|S|}$ where the sum is over all dominating sets $S$. We prove that $P_{m \times n}(-1) = (-1)^{\lceil m/2\rceil \lceil n/2\rceil}$. In particular, the number of dominating sets of even size and the number of odd size differs by $\pm 1$. %The numbers can not be equal because the total number of dominating sets is always odd. This property does not hold for king graphs on a cylinder or a torus, or for the grid graph. But it holds for $d$-dimensional kings, where $P_{n_1\times n_2\times\cdots\times n_d}(-1) = (-1)^{\lceil n_1/2\rceil \lceil n_2/2\rceil\cdots \lceil n_d/2\rceil}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Domination by kings is oddly even Moore, Cristopher Mertens, Stephan Combinatorics 05C69, 05A15 The $m \times n$ king graph consists of all locations on an $m \times n$ chessboard, where edges are legal moves of a chess king. %where each vertex represents a square on a chessboard and each edge is a legal move. Let $P_{m \times n}(z)$ denote its domination polynomial, i.e., $\sum_{S \subseteq V} z^{|S|}$ where the sum is over all dominating sets $S$. We prove that $P_{m \times n}(-1) = (-1)^{\lceil m/2\rceil \lceil n/2\rceil}$. In particular, the number of dominating sets of even size and the number of odd size differs by $\pm 1$. %The numbers can not be equal because the total number of dominating sets is always odd. This property does not hold for king graphs on a cylinder or a torus, or for the grid graph. But it holds for $d$-dimensional kings, where $P_{n_1\times n_2\times\cdots\times n_d}(-1) = (-1)^{\lceil n_1/2\rceil \lceil n_2/2\rceil\cdots \lceil n_d/2\rceil}$. |
| title | Domination by kings is oddly even |
| topic | Combinatorics 05C69, 05A15 |
| url | https://arxiv.org/abs/2407.19344 |