On variants of conflict-free-coloring for hypergraphs

Conflict-free coloring is a kind of vertex coloring of hypergraphs requiring each hyperedge to have a color which appears only on one vertex. More generally, for a positive integer k there are k-conflict-free colorings (k-CF-colorings for short) and k-strong-conflict-free colorings (k-SCF-colorings for short). Let Hn be the hypergraph of which the vertex-set is Vn={1,2,…,n} and the hyperedge-set En is the set of all (non-empty) subsets of Vn consisting of consecutive elements of Vn. Firstly, we study the k-SCF-coloring of Hn, give the exact k-SCF-coloring chromatic number of Hn for k=2,3, and present upper and lower bounds of the k-SCF-coloring chromatic number of Hn for all k. Secondly, we give the exact k-CF-coloring chromatic number of Hn for all k.