An improved upper bound on the density of universal random graphs2. Issue 2 (22nd May 2014)

• Record Type: Journal Article
• Authors: Dellamonica, Domingos; Kohayakawa, Yoshiharu; Rödl, Vojtěch; Ruciński, Andrzej
• Is Part Of: Random structures & algorithms. Volume 46:Issue 2(2015)
• Page Start: 274
• Page End: 299
• Publication Date: 2014-05-22
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20545
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We give a polynomial time randomized algorithm that, on receiving as input a pair (<italic>H, G</italic>) of <italic>n</italic>‐vertex graphs, searches for an embedding of <italic>H</italic> into <italic>G</italic>. If <italic>H</italic> has bounded maximum degree and <italic>G</italic> is suitably dense and pseudorandom, then the algorithm succeeds with high probability. Our algorithm proves that, for every integer <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734f1bb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20545:rsa20545-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> and a large enough constant <italic>C</italic> = <italic>C</italic><sub><italic>d</italic></sub>, as <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734f19s" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20545:rsa20545-math-0002" 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>, asymptotically almost all graphs with <italic>n</italic> vertices and at least <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734f187" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20545:rsa20545-math-0003" 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>2</mml:mn><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>log</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> edges contain as subgraphs all graphs with <italic>n</italic> vertices and maximum degree at most <italic>d</italic>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 2014</p> </abstract>