On a greedy 2‐matching algorithm and Hamilton cycles in random graphs with minimum degree at least three1. Issue 3 (27th December 2012)

• Record Type: Journal Article
• Authors: Frieze, Alan
• Is Part Of: Random structures & algorithms. Volume 45:Issue 3(2014)
• Page Start: 443
• Page End: 497
• Publication Date: 2012-12-27
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20482
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>We describe and analyse a simple greedy algorithm 2greedy that finds a good 2‐matching <italic>M</italic> in the random graph <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v92599" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20482:rsa20482-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mi>G</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>δ</mml:mo><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula> when <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9258r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20482:rsa20482-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mo>≥</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. A 2‐matching is a spanning subgraph of maximum degree two and <italic>G</italic> is drawn uniformly from graphs with vertex set <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v92576" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20482:rsa20482-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, <italic>cn</italic> edges and minimum degree at least three. By good we mean that <italic>M</italic> has <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9256n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20482:rsa20482-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> components. We then use this 2‐matching to build a Hamilton cycle in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v92553" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20482:rsa20482-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>1.5</mml:mn><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:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> time w.h.p. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 443‐497, 2014</p> </abstract>