A note on random greedy coloring of uniform hypergraphs. Issue 3 (30th May 2014)

• Record Type: Journal Article
• Authors: Cherkashin, Danila D.; Kozik, Jakub
• Is Part Of: Random structures & algorithms. Volume 47:Issue 3(2015)
• Page Start: 407
• Page End: 413
• Publication Date: 2014-05-30
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20556
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The smallest number of edges forming an <italic>n</italic>‐uniform hypergraph which is not <italic>r</italic>‐colorable is denoted by <italic>m</italic>(<italic>n, r</italic>). Erdős and Lovász conjectured that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9tsnk1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20556:rsa20556-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>Θ</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>. The best known lower bound <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9tsnrs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20556:rsa20556-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>Ω</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msqrt><mml:mrow><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msqrt><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on the analysis of a random greedy coloring algorithm investigated by Pluhár in 2009. The proof method extends to the case of <italic>r</italic>‐coloring, and we show that for any fixed <italic>r</italic> we have <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9tsnpp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20556:rsa20556-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>Ω</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mo> </mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> improving the bound of Kostochka from 2004. We also derive analogous bounds on minimum edge degree of an <italic>n</italic>‐uniform hypergraph that is not <italic>r</italic>‐colorable. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 407–413, 2015</p> </abstract>