Harmonious and achromatic colorings of fragmentable hypergraphs

A harmonious coloring of a k-uniform hypergraph H 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 number (resp. achromatic number) of H, denoted by h(H) (resp. psi(H)) is the minimum (resp. maximum) possible number of colors in a harmonious (resp. achromatic) coloring of H. A class of hypergraphs is fragmentable if for every H is an element of H, H can be fragmented into 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 is an element of > 0 and for every hypergraph H is an element of H with m >= m(0)(H, is an element of) edges we have h(H) <= (1 + is an element of)(k)root k!m and psi(H) >= (1 - is an element of)(k)root k!m. As corollaries, we answer a question posed by Blackburn concerning the maximum length of t-subset packing 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 Surds and Schelp. (C) 2017 Elsevier Ltd. All rights reserved.