$2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911578528415744 |
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| author | Mondal, Anupam Mukherjee, Sajal Pramanik, Pritam Chandra |
| author_facet | Mondal, Anupam Mukherjee, Sajal Pramanik, Pritam Chandra |
| contents | We obtain a higher dimensional analogue of a classical theorem which states that a polygonally cellulated $2$-sphere in $\mathbb{R}^3$, such that each vertex has even degree, is $2$-face-colourable. In order to formulate our result, we introduce the notion of combinatorially sphere-like ranked posets, which are ranked posets that generalise combinatorial spheres. We prove that, in a combinatorially sphere-like ranked poset $S$ of rank $k$, if each element of rank $(k-2)$ is covered by an even number of elements, then the maximum ranked elements of $S$ admit a proper $2$-colouring, i.e., any two adjacent maximum ranked elements have different colours. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08210 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset Mondal, Anupam Mukherjee, Sajal Pramanik, Pritam Chandra Combinatorics 06A07, 05C15, 52B05 We obtain a higher dimensional analogue of a classical theorem which states that a polygonally cellulated $2$-sphere in $\mathbb{R}^3$, such that each vertex has even degree, is $2$-face-colourable. In order to formulate our result, we introduce the notion of combinatorially sphere-like ranked posets, which are ranked posets that generalise combinatorial spheres. We prove that, in a combinatorially sphere-like ranked poset $S$ of rank $k$, if each element of rank $(k-2)$ is covered by an even number of elements, then the maximum ranked elements of $S$ admit a proper $2$-colouring, i.e., any two adjacent maximum ranked elements have different colours. |
| title | $2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset |
| topic | Combinatorics 06A07, 05C15, 52B05 |
| url | https://arxiv.org/abs/2604.08210 |