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arXiv · 2408.09391

On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs

Seunghun Lee·Shakhar Smorodinsky
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Abstract

We study the conflict-free chromatic number of hypergraphs derived from the family of facets of $d$-dimensional cyclic polytopes with $n$ vertices. While in odd dimensions $d$ the problem is easy, for even dimensions the problem becomes very difficult and exhibits interesting connections to extremal graph theory. We provide sharp asymptotic bounds for the conflict-free chromatic number in several small even dimensions and non-trivial upper and lower bounds for general even dimensions. The main purpose of this paper is revealing a surprising relation between conflict-free colorings and the celebrated Erd\H{o}s girth conjecture, opening new avenues for future research.

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Seunghun Lee, Shakhar Smorodinsky. 2024-08-18. On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs. https://arxiv.org/abs/2408.09391

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