The thresholds for diameter 2 in random Cayley graphs1. Issue 2 (5th March 2013)

• Record Type: Journal Article
• Authors: Christofides, Demetres; Markström, Klas
• Is Part Of: Random structures & algorithms. Volume 45:Issue 2(2014)
• Page Start: 218
• Page End: 235
• Publication Date: 2013-03-05
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20486
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>Given a group <italic>G</italic>, the model <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p0nfm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20486:rsa20486-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo>, </mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> denotes the probability space of all Cayley graphs of <italic>G</italic> where each element of the generating set is chosen independently at random with probability <italic>p</italic>.</p> <p>In this article we show that for any <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p0n7h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20486:rsa20486-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ε</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> and any family of groups <italic>G</italic><sub><italic>k</italic></sub> of order <italic>n</italic><sub><italic>k</italic></sub> for which <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p0n81" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20486:rsa20486-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, a graph <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p0pdk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20486:rsa20486-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mo>Γ</mml:mo><mml:mi>k</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>, </mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> with high probability has diameter at most 2 if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p0pbj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20486:rsa20486-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:msqrt><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mo>ε</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:mi>log</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula> and with high probability has diameter greater than 2 if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p0p91" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20486:rsa20486-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:msqrt><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mo>ε</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>log</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula>.</p> <p>We also provide examples of families of graphs which show that both of these results are best possible.</p> <p>Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erdős‐Renyi random graphs. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 218–235, 2014</p> </abstract>