An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three1. Issue 1 (10th April 2014)

• Record Type: Journal Article
• Authors: Frieze, Alan; Haber, Simi
• Is Part Of: Random structures & algorithms. Volume 47:Issue 1(2015)
• Page Start: 73
• Page End: 98
• Publication Date: 2014-04-10
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20542
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj14cqs7bp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20542:rsa20542-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>m</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>. In this model <italic>G</italic> is drawn uniformly from graphs with vertex set [<italic>n</italic>], <italic>m</italic> edges and minimum degree at least three. We focus on the case where <italic>m</italic> = <italic>cn</italic> for constant <italic>c</italic>. If <italic>c</italic> is sufficiently large then our algorithm runs in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj14cqs794" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20542:rsa20542-math-0002" 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</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 and succeeds w.h.p. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 73–98, 2015</p> </abstract>