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

TL;DR

This paper explores the relationship between the chromatic numbers of hypergraphs and their 1-intersection graphs, proving colorability results for specific intersection graph structures.

Contribution

It establishes that hypergraphs with 1-intersection graphs that are 2- or 4-partite can be properly colored with 2 or 4 colors, respectively.

Findings

Hypergraphs with 1-intersection graphs that are 2-partite are 2-colorable.

Hypergraphs with 1-intersection graphs that are 4-partite are 4-colorable.

Extends classical coloring results to hypergraphs via intersection graph properties.

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.