| Tartalmi leírás: |
The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets.
We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called (em ABA-free hypergraphs), which are a special case of (em shift-chains).
Using our methods, we prove that $(2k-1)$-uniform ABA-free hypergraphs have a polychromatic $k$-coloring, a problem posed by the second author.
We also prove the same for hypergraphs defined on a point set by pseudohalfplanes.
These results are best possible.
We also introduce several new notions that seem to be important for investigating polychromatic colorings and $epsilon$-nets, such as (em shallow hitting sets) and (em balanced polychromatic colorings).
We pose several open problems related to them.
For example, is it true that given a finite point set $S$ on a sphere and a set of halfspheres $FF$, such that $(Scap Fmid Fin FF)$ is a Sperner family, we can select an $Rsubset S$ such that $1le |Fcap R|le 2$ holds for every $Fin FF$?
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| Azonosító: |
Keszegh, Balázs and Pálvölgyi, Dömötör (2015) An abstract approach to polychromatic coloring: shallow hitting sets in ABA-free hypergraphs and pseudohalfplanes. In: 21st International Workshop on Graph-Theoretic Concepts in Computer Science, 2015. June 17-19., Munich, Germany. (In Press)
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