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 On variants of conflict-free-coloring for hypergraphs


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Conflict-free coloring is a kind of vertex coloring of hypergraphs requiring each hyperedge to have a color which appears only once. More generally, for a positive integer $k$ there are $k$-conflict-free coloring $k$-CF-coloring for short and $k$-strong-conflict-free coloring $k$-SCF-coloring for short for some positive integer $k$. Let $H n$ be the hypergraph of which the vertex-set $V n=\{1,2,\dots,n\}$ and the hyperedge-set $\cal{E} n$ is the set of all non-empty subsets of $V n$ consisting of consecutive elements of $V n$. Firstly, we study the $k$-SCF-coloring of $H n$, give the exact $k$-SCF-coloring number of $H n$ for $k=2,3$, and for arbitrary $k$ present upper and lower bounds of the $k$-SCF-coloring number of $H n$ for all $k$. Secondly, we give the exact $k$-CF-coloring number of $H n$ for all $k$. Finally, we extend some results about online conflict-free coloring for hypergraphs obtained in \cite{8} to online $k$-CF-coloring.



Author: Zhen Cui; Ze-Chun Hu

Source: https://archive.org/







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