Tight Hamilton cycles in random hypergraphs. Issue 3 (16th October 2013)

• Record Type: Journal Article
• Authors: Allen, Peter; Böttcher, Julia; Kohayakawa, Yoshiharu; Person, Yury
• Is Part Of: Random structures & algorithms. Volume 46:Issue 3(2015)
• Page Start: 446
• Page End: 465
• Publication Date: 2013-10-16
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20519
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We give an algorithmic proof for the existence of tight Hamilton cycles in a random <italic>r</italic>‐uniform hypergraph with edge probability <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82hr1s" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20519:rsa20519-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>ε</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> for every <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82hr29" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20519:rsa20519-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>. This partly answers a question of Dudek and Frieze (Random Struct Algor 42 (2013), 374–385), who used a second moment method to show that tight Hamilton cycles exist even for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82hqw7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20519:rsa20519-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mo>ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo> </mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> where <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82hqz8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20519:rsa20519-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>→</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> arbitrary slowly, and for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82hqsp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20519:rsa20519-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:mo stretchy="false">(</mml:mo><mml:mi>e</mml:mi><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo> </mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. The method we develop for proving our result applies to related problems as well. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 446–465, 2015</p> </abstract>