Edge‐disjoint Hamilton cycles in random graphs1. Issue 3 (2nd July 2013)

• Record Type: Journal Article
• Authors: Knox, Fiachra; Kühn, Daniela; Osthus, Deryk
• Is Part Of: Random structures & algorithms. Volume 46:Issue 3(2015)
• Page Start: 397
• Page End: 445
• Publication Date: 2013-07-02
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20510
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We show that provided <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82pdsk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20510:rsa20510-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mi>log</mml:mi></mml:mrow><mml:mrow><mml:mn>50</mml:mn></mml:mrow></mml:msup><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><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:mn>4</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>log</mml:mi></mml:mrow><mml:mn>9</mml:mn></mml:msup><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> we can with high probability find a collection of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82pf3q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20510:rsa20510-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>⌊</mml:mo><mml:mrow><mml:mo>δ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>⌋</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> edge‐disjoint Hamilton cycles in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82pf26" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20510:rsa20510-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>∼</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>, plus an additional edge‐disjoint matching of size <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82pf1p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20510:rsa20510-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>⌊</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>⌋</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82pdxn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20510:rsa20510-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>δ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> is odd. This is clearly optimal and confirms, for the above range of <italic>p</italic>, a conjecture of Frieze and Krivelevich. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 397–445, 2015</p> </abstract>