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arXiv · 1207.4415

Online and quasi-online colorings of wedges and intervals

Balázs Keszegh·Nathan Lemons·Dömötör Pálvölgyi
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Abstract

We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring algorithm that colors $N$ intervals of the real line using $Θ(\log N/k)$ colors such that for every point $p$, contained in at least $k$ intervals, not all the intervals containing $p$ have the same color. We also prove the corresponding result about online coloring a family of wedges (quadrants) in the plane that are the translates of a given fixed wedge. These results contrast the results of the first and third author showing that in the quasi-online setting 12 colors are enough to color wedges (independent of $N$ and $k$). We also consider quasi-online coloring of intervals. In all cases we present efficient coloring algorithms.

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Balázs Keszegh, Nathan Lemons, Dömötör Pálvölgyi. 2015-09-09. Online and quasi-online colorings of wedges and intervals. https://arxiv.org/abs/1207.4415

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