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

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 ∈ { 2 , 4 } that all hypergraphs whose 1-intersection graph is k -partite can be properly k -colored.