The Cℓ‐free process. Issue 4 (6th November 2012)

• Record Type: Journal Article
• Authors: Warnke, Lutz
• Is Part Of: Random structures & algorithms. Volume 44:Issue 4(2014)
• Page Start: 490
• Page End: 526
• Publication Date: 2012-11-06
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20468
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fmnj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20468:rsa20468-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mo>ℓ</mml:mo></mml:msub></mml:mrow></mml:math></alternatives>‐free process starts with the empty graph on <bold><italic>n</italic></bold> vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fmm1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20468:rsa20468-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mo>ℓ</mml:mo></mml:msub></mml:mrow></mml:math></alternatives> is created. For every <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fmsm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20468:rsa20468-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ℓ</mml:mo><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></alternatives> we show that, with high probability as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fmr3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20468:rsa20468-math-0005" 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 is <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fmqk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20468:rsa20468-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>ℓ</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>, which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fmp2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20468:rsa20468-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mo>ℓ</mml:mo></mml:msub></mml:mrow></mml:math></alternatives>‐free process typically terminates with <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fmvn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20468:rsa20468-math-0008" 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:mo>ℓ</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>ℓ</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></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>1</mml:mn><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>ℓ</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> edges, which answers a question of Erdős, Suen and Winkler. This is the first result that determines the final number of edges of the more general <italic>H</italic>‐free process for a non‐trivial <italic>class</italic> of graphs <italic>H</italic>. We also verify a conjecture of Osthus and Taraz concerning the average degree, and obtain a new lower bound on the independence number. Our proof combines the differential equation method with a tool that might be of independent interest: we establish a rigorous way to 'transfer' certain decreasing properties from the binomial random graph to the <italic>H</italic>‐free process. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 490–526, 2014</p> </abstract>