The connection between the chromatic numbers of a hypergraph and its $1$-intersection graph
Abstract
A well known problem from an excellent book of Lovász 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 Gyárfás 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.12118
- arXiv:
- arXiv:2406.12118
- Bibcode:
- 2024arXiv240612118B
- Keywords:
-
- Mathematics - Combinatorics;
- 05C15