When does the K4‐free process stop?. Issue 3 (27th July 2012)

• Record Type: Journal Article
• Authors: Warnke, Lutz
• Is Part Of: Random structures & algorithms. Volume 44:Issue 3(2014)
• Page Start: 355
• Page End: 397
• Publication Date: 2012-07-27
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20444
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The <italic>K</italic><sub>4</sub>‐free process starts with the empty graph on <italic>n</italic> vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of <italic>K</italic><sub>4</sub>. Let <italic>G</italic> be the random maximal <italic>K</italic><sub>4</sub>‐free graph obtained at the end of the process. We show that for some positive constant <italic>C</italic>, with high probability as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1f1d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20444:rsa20444-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives>, the maximum degree in <italic>G</italic> is at most <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1ff1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20444:rsa20444-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mroot><mml:mrow><mml:mi>log</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:mroot></mml:mrow></mml:math></alternatives>. This resolves a conjecture of Bohman and Keevash for the <italic>K</italic><sub>4</sub>‐free process and improves on previous bounds obtained by Bollobás and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability <italic>G</italic> has <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1fdg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20444:rsa20444-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Θ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>8</mml:mn><mml:mo>/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mroot><mml:mrow><mml:mi>log</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:mroot><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> edges and is 'nearly regular', i.e., every vertex has degree <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1fms" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20444:rsa20444-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Θ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mroot><mml:mrow><mml:mi>log</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:mroot><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>. This answers a question of Erdős, Suen and Winkler for the <italic>K</italic><sub>4</sub>‐free process. We furthermore deduce an additional structural property: we show that whp the independence number of <italic>G</italic> is at least <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1fh4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20444:rsa20444-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo>/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mi>log</mml:mi><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>, which matches an upper bound obtained by Bohman up to a factor of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1fqf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20444:rsa20444-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Θ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>. Our analysis of the <italic>K</italic><sub>4</sub>‐free process also yields a new result in Ramsey theory: for a special case of a well‐studied function introduced by Erdős and Rogers we slightly improve the best known upper bound.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 355‐397, 2014</p> </abstract>