Coloring one-headed directed hypergraphs

A directed hypergraph is a hypergraph in which the vertex set of each hyperedge is partitioned into two disjoint parts, a head and a tail. Keszegh and P\'alv\"olgyi posed the following conjecture. Let H be a directed hypergraph such that in every hyperedge the number of head-vertices is less than the number of tail-vertices and assume that for every pair of hyperedges e₁,e₂∈ E(H) with |e₁∩ e₂|=1, the common vertex is a head-vertex in at least one of the hyperedges. Then H admits a proper 2-coloring. Keszegh showed that the conjecture is also true in the special case of 3-uniform hypergraphs. A directed hypergraph is called one-headed if every hyperedge has exactly one head-vertex. The main result of this paper is that the conjecture is true for one-headed directed hypergraphs with all hyperedges having size at least three. Directed 3-uniform hypergraphs such that in every hyperedge the number of head-vertices is one and the number of tail-vertices is two are called 2→ 1 hypergraphs. In this paper we consider sufficient conditions for 2→ 1 hypergraphs to be proper k-colorable for some small k.