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arXiv · 2406.12118

The connection between the chromatic numbers of a hypergraph and its $1$-intersection graph

Zoltán L. Blázsik·Nathan W. Lemons
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Abstract

A well known problem from an excellent book of Lov\'asz states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh and of Gy\'arf\'as et al we study the $1$-intersection graph of a hypergraph. The $1$-intersection graph encodes those pairs of hyperedges in a hypergraph that intersect in exactly one vertex. We prove for $k\in\{2,4\}$ that all hypergraphs whose $1$-intersection graph is $k$-partite can be properly $k$-colored.

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Zoltán L. Blázsik, Nathan W. Lemons. 2024-06-17. The connection between the chromatic numbers of a hypergraph and its $1$-intersection graph. https://arxiv.org/abs/2406.12118

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