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arXiv · 2604.08210

$2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset

Anupam Mondal·Sajal Mukherjee·Pritam Chandra Pramanik
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Abstract

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.

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Anupam Mondal, Sajal Mukherjee, Pritam Chandra Pramanik. 2026-04-09. $2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset. https://arxiv.org/abs/2604.08210

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