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

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.