Cover time of a random graph with a degree sequence II: Allowing vertices of degree two1. Issue 4 (4th October 2014)

• Record Type: Journal Article
• Authors: Cooper, Colin; Frieze, Alan; Lubetzky, Eyal
• Is Part Of: Random structures & algorithms. Volume 45:Issue 4(2014)
• Page Start: 627
• Page End: 674
• Publication Date: 2014-10-04
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20573
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We study the cover time of a random graph chosen uniformly at random from the set of graphs with vertex set [<italic>n</italic>] and degree sequence <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh21kb0dnk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20573:rsa20573-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="bold">d</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>. In a previous work (Abdullah, Cooper, and Frieze, Discrete Math 312 (2012), 3146–3163), the asymptotic cover time was obtained under a number of assumptions on <bold>d</bold>, the most significant being that <italic>d</italic><sub><italic>i</italic></sub> ≥ 3 for all <italic>i</italic>. Here we replace this assumption by <italic>d</italic><sub><italic>i</italic></sub> ≥ 2. As a corollary, we establish the asymptotic cover time for the 2‐core of the emerging giant component of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh21kb0dp4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20573:rsa20573-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 627–674, 2014</p> </abstract>