The diamond‐free process. Issue 3 (11th October 2013)

• Record Type: Journal Article
• Authors: Picollelli, Michael E.
• Is Part Of: Random structures & algorithms. Volume 45:Issue 3(2014)
• Page Start: 513
• Page End: 551
• Publication Date: 2013-10-11
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20517
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Let <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9bmjr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20517:rsa20517-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mn>4</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula> denote the diamond graph, formed by removing an edge from the complete graph <italic>K</italic><sub>4</sub>. We consider the following random graph process: starting with <italic>n</italic> isolated vertices, add edges uniformly at random provided no such edge creates a copy of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9bmk9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20517:rsa20517-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mn>4</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>. We show that, with probability tending to 1 as <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9bm3k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20517:rsa20517-math-0003" 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></inline-formula>, the final size of the graph produced is <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9bm7s" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20517:rsa20517-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Θ</mml:mo><mml:mo>(</mml:mo><mml:msqrt><mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msqrt><mml:mo>·</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Our analysis also suggests that the graph produced after <italic>i</italic> edges are added resembles the uniform random graph, with the additional condition that the edges which do not lie on triangles form a random‐looking subgraph. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 513–551, 2014</p> </abstract>