On variants of conflict-free-coloring for hypergraphs

الأصل · EN

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). %for some positive integer k. 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.