On covering expander graphs by hamilton cycles. Issue 2 (10th August 2012)

• Record Type: Journal Article
• Authors: Glebov, Roman; Krivelevich, Michael; Szabó, Tibor
• Is Part Of: Random structures & algorithms. Volume 44:Issue 2(2014)
• Page Start: 183
• Page End: 200
• Publication Date: 2012-08-10
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20455
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree Δ satisfies some basic expansion properties and contains a family of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2rngd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20455:rsa20455-math-0002" 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:mo>Δ</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives> edge disjoint Hamilton cycles, then there also exists a covering of its edges by <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2rnhz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20455:rsa20455-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:mo>Δ</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives> Hamilton cycles. This implies that for every α > 0 and every <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2rnjh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20455:rsa20455-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:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>α</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives> there exists a covering of all edges of <italic>G</italic>(<italic>n, p</italic>) by <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2rnk2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20455:rsa20455-math-0005" 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>n</mml:mi><mml:mi>p</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives> Hamilton cycles asymptotically almost surely, which is nearly optimal.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 183‐200, 2014</p> </abstract>