Harmonious and achromatic colorings of fragmentable hypergraphs

A harmonious coloring of a k-uniform hypergraphH is a rainbow vertex coloring such that each k-set of colors appears on at most one edge. A rainbow coloring of H is achromatic if each k-set of colors appears on at least one edge. The harmonious (resp. achromatic) number of H , denoted by h(H) (resp. ψ(H)) is the minimum (resp. maximum) possible number of colors in a harmonious (resp. achromatic) coloring of H . A class H of hypergraphs is fragmentable if for every H ∈ H, H can be fragmented to components of a bounded size by removing a „small” fraction of vertices. We show that for every fragmentable class H of bounded degree hypergraphs, for every > 0 and for every hypergraph H ∈ H with m ≥ m0(H, ) edges we have h(H) ≤ (1 + ) k √ k!m and ψ(H) ≥ (1− ) k √k!m. As corollaries, we answer a question posed by Blackburn (concerning the maximum length of packing t-subset sequences of constant radius) and derive an asymptotically tight bound on the minimum number of colors in a vertex-distinguishing edge coloring of cubic planar graphs (which is a step towards confirming a conjecture of Burris and Schelp).