Invariant Gaussian processes and independent sets on regular graphs of large girth1. Issue 2 (16th May 2014)

• Record Type: Journal Article
• Authors: Csóka, Endre; Gerencsér, Balázs; Harangi, Viktor; Virág, Bálint
• Is Part Of: Random structures & algorithms. Volume 47:Issue 2(2015)
• Page Start: 284
• Page End: 303
• Publication Date: 2014-05-16
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20547
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We prove that every 3‐regular, <italic>n</italic>‐vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361<italic>n</italic>. (The best known bound is 0.4352<italic>n</italic>.) In fact, computer simulation suggests that the bound our method provides is about 0.438<italic>n</italic>.</p> <p>Our method uses invariant Gaussian processes on the <italic>d</italic>‐regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twhtsc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20547:rsa20547-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>λ</mml:mo></mml:math></alternatives></inline-formula>. We show that such processes can be approximated by i.i.d. factors provided that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twhttx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20547:rsa20547-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mo>λ</mml:mo><mml:mo>|</mml:mo><mml:mo>≤</mml:mo><mml:mn>2</mml:mn><mml:msqrt><mml:mrow><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula>. We then use these approximations for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twhtvg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20547:rsa20547-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>λ</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msqrt><mml:mrow><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula> to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 284–303, 2015</p> </abstract>