Long paths and cycles in random subgraphs of graphs with large minimum degree1. Issue 2 (27th June 2013)

• Record Type: Journal Article
• Authors: Krivelevich, Michael; Lee, Choongbum; Sudakov, Benny
• Is Part Of: Random structures & algorithms. Volume 46:Issue 2(2015)
• Page Start: 320
• Page End: 345
• Publication Date: 2013-06-27
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20508
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>For a given finite graph <italic>G</italic> of minimum degree at least <italic>k</italic>, let <italic>G</italic><sub><italic>p</italic></sub> be a random subgraph of <italic>G</italic> obtained by taking each edge independently with probability <italic>p</italic>. We prove that (i) if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734k4g1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20508:rsa20508-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:mo>ω</mml:mo><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> for a function <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734k4cc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20508:rsa20508-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ω</mml:mo><mml:mo>=</mml:mo><mml:mo>ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> that tends to infinity as <italic>k</italic> does, then <italic>G</italic><sub><italic>p</italic></sub> asymptotically almost surely contains a cycle (and thus a path) of length at least <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734k498" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20508:rsa20508-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><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:mo stretchy="false">)</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>, and (ii) if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734k48q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20508:rsa20508-math-0004" 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: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:mo stretchy="false">)</mml:mo><mml:mi>ln</mml:mi><mml:mi>k</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>, then <italic>G</italic><sub><italic>p</italic></sub> asymptotically almost surely contains a path of length at least <italic>k</italic>. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking <italic>G</italic> to be the complete graph on <italic>k</italic> + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015</p> </abstract>