Generating random graphs in biased Maker‐Breaker games1. Issue 4 (12th October 2015)

• Record Type: Journal Article
• Authors: Ferber, Asaf; Krivelevich, Michael; Naves, Humberto
• Is Part Of: Random structures & algorithms. Volume 47:Issue 4(2015)
• Page Start: 615
• Page End: 634
• Publication Date: 2015-10-12
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20619
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We present a general approach connecting biased Maker‐Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker‐Breaker games. In particular, we show that for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvp14k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20619:rsa20619-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mi>o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvp195" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20619:rsa20619-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>b</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> game on <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvp18n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20619:rsa20619-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. As another application, we show that for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvp1js" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20619:rsa20619-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mo>Θ</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>ln</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>, playing a <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvp1dq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20619:rsa20619-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>b</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> game on <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvp1sd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20619:rsa20619-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, Maker can build a graph which contains copies of all spanning trees having maximum degree <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvp1mt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20619:rsa20619-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Δ</mml:mo><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:math></alternatives></inline-formula> with a bare path of linear length (a bare path in a tree <italic>T</italic> is a path with all interior vertices of degree exactly two in <italic>T</italic>). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 47, 615–634, 2015</p> </abstract>