License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.46
URN: urn:nbn:de:0030-drops-90505
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9050/
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Duraj, Lech ; Gutowski, Grzegorz ; Kozik, Jakub

A Note on Two-Colorability of Nonuniform Hypergraphs

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Abstract

For a hypergraph H, let q(H) denote the expected number of monochromatic edges when the color of each vertex in H is sampled uniformly at random from the set of size 2. Let s_{min}(H) denote the minimum size of an edge in H. Erdös asked in 1963 whether there exists an unbounded function g(k) such that any hypergraph H with s_{min}(H) >=slant k and q(H) <=slant g(k) is two colorable. Beck in 1978 answered this question in the affirmative for a function g(k) = Theta(log^* k). We improve this result by showing that, for an absolute constant delta>0, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph H with s_{min}(H) >=slant k and q(H) <=slant delta * log k.

BibTeX - Entry

@InProceedings{duraj_et_al:LIPIcs:2018:9050,
  author =	{Lech Duraj and Grzegorz Gutowski and Jakub Kozik},
  title =	{{A Note on Two-Colorability of Nonuniform Hypergraphs}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{46:1--46:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9050},
  URN =		{urn:nbn:de:0030-drops-90505},
  doi =		{10.4230/LIPIcs.ICALP.2018.46},
  annote =	{Keywords: Property B, Nonuniform Hypergraphs, Hypergraph Coloring, Random Greedy Coloring}
}

Keywords: Property B, Nonuniform Hypergraphs, Hypergraph Coloring, Random Greedy Coloring
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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