  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.
