Color-bounded hypergraphs, V: host graphs and subdivisions

Abstract

A color-bounded hypergraph is a hypergraph (set system) with ver- tex set X and edge set e = {E1, . . . ,Em}, together with integers si and ti satisfying 1 ≤ si ≤ ti ≤ |E1| for each i = 1, . . . ,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge 1 satisfies si ≤ |φ(Ei)| ≤ t i. The hypergraph H is colorable if it admits at least one proper coloring. We consider hypergraphs H over a graph, that means a graph G on the same vertex set X as H, such that each 1 induces a connected subgraph in G. In the current setting we fix a graph or multigraph G0, and assume that the host graph G is obtained by some sequence of edge subdivisions, starting from G0. The colorability problem is known to be NP-complete in general, and also when restricted to 3-uniform mixed hypergraphs, i.e., color- bounded hypergraphs in which |1| = 3 and 1 ≤ si ≤ 2 ≤ ti ≤ 3 holds for all i ≤ m. We prove that for every fixed graph G0 and natural number r, colorability is decidable in polynomial time over the class of r-uniform hypergraphs (and more generally of hypergraphs with |1| ≤ r for all 1 ≤ i ≤ m) having a host graph G obtained from G0 by edge subdivisions. Stronger bounds are derived for hypergraphs for which G0 is a tree.