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**1 - 3**of**3**### Conflict-free coloring of graphs

, 2013

"... We study the conflict-free chromatic number χCF of graphs from ex-tremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution o ..."

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We study the conflict-free chromatic number χCF of graphs from ex-tremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős-Rényi random graph G(n, p) and give the asymptotics for p = ω(1/n). We also show that for p ≥ 1/2 the conflict-free chromatic number differs from the domination number by at most 3. MSC classes: 05C35, 05C15, 05C80, 05D40, 05C69. 1 Introduction and

### CONFLICT-FREE COLORINGS OF UNIFORM HYPERGRAPHS WITH FEW EDGES

, 2012

"... A coloring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose color does not repeat in e. The smallest number of colors required for such a coloring is called the conflict-free chromatic number of H, and is denoted by χCF (H). Pach and Tardos prov ..."

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A coloring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose color does not repeat in e. The smallest number of colors required for such a coloring is called the conflict-free chromatic number of H, and is denoted by χCF (H). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph H with m edges, χCF (H) is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colorable.