Geodesic
Color-bounded hypergraphs, V: host graphs and subdivisions
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 223-238

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A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = E₁,...,Eₘ, together with integers s_i and t_i satisfying 1 ≤ s_i ≤ t_i ≤ |E_i| for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge E_i satisfies s_i ≤ |φ(E_i)| ≤ t_i. The hypergraph ℋ is colorable if it admits at least one proper coloring.
Keywords: mixed hypergraph, color-bounded hypergraph, vertex coloring, arboreal hypergraph, hypertree, feasible set, host graph, edge subdivision 
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Bujtás, Csilla; Tuza, Zsolt; Voloshin, Vitaly. Color-bounded hypergraphs, V: host graphs and subdivisions. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 223-238. http://geodesic.mathdoc.fr/item/DMGT_2011_31_2_a1/