Sandwiching a densest subgraph by consecutive cores1. Issue 2 (10th April 2014)

• Record Type: Journal Article
• Authors: Gao, Pu
• Is Part Of: Random structures & algorithms. Volume 47:Issue 2(2015)
• Page Start: 341
• Page End: 360
• Publication Date: 2014-04-10
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20543
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>In this paper, we show that in the random graph <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh1h2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0001" 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>c</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, with high probability, there exists an integer <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh1gh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:math></alternatives></inline-formula> such that a subgraph of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh1fz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0003" 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>c</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, whose vertex set differs from a densest subgraph of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh1dd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0004" 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>c</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> by <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh1cv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-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:mrow><mml:mi>log</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> vertices, is sandwiched by the <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh1b9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:math></alternatives></inline-formula> and the <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh19r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo>˜</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>‐core, for almost all sufficiently large <italic>c</italic>. We determine the value of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh186" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:math></alternatives></inline-formula>. We also prove that (a), the threshold of the <italic>k</italic>‐core being balanced coincides with the threshold that the average degree of the <italic>k</italic>‐core is at most <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh17n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, for all sufficiently large <italic>k</italic>; (b) with high probability, there is a subgraph of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twh0sz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20543:rsa20543-math-0010" 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>c</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> whose density is significantly denser than any of its nonempty cores, for almost all sufficiently large <italic>c</italic> > 0. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 341–360, 2015</p> </abstract>