Unique-Maximum and Conflict-Free Coloring for Hypergraphs and Tree Graphs
We investigate the relationship between two kinds of vertex colorings of hypergraphs: unique-maximum colorings and conflict-free colorings. In a unique-maximum coloring, the colors are ordered, and in every hyperedge of the hypergraph the maximum color in the hyperedge occurs in only one vertex of the hyperedge. In a conflict-free coloring, in every hyperedge of the hypergraph there exists a color in the hyperedge that occurs in only one vertex of the hyperedge. We define corresponding unique-maximum and conflict-free chromatic numbers and investigate their relationship in arbitrary hypergraphs. Then, we concentrate on hypergraphs that are induced by simple paths in tree graphs.
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