Biased games on random boards1. Issue 4 (24th February 2014)

• Record Type: Journal Article
• Authors: Ferber, Asaf; Glebov, Roman; Krivelevich, Michael; Naor, Alon
• Is Part Of: Random structures & algorithms. Volume 46:Issue 4(2015)
• Page Start: 651
• Page End: 676
• Publication Date: 2014-02-24
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20528
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>In this paper we analyze biased Maker‐Breaker games and Avoider‐Enforcer games, both played on the edge set of a random board <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kdmp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>∼</mml:mo><mml:mi>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>. In Maker‐Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of G and Breaker responds by claiming b previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the k‐vertex‐connectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider‐Enforcer games are the reverse analogue of Maker‐Breaker games with a slight modification, where the two players claim at least 1 and at least b previously unclaimed edges per move, respectively, and Avoider aims to avoid building a graph which possesses the relevant property.</p> <p>Maker‐Breaker games are known to be "bias‐monotone", that is, if Maker wins the (1, <italic>b</italic>) game, he also wins the <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kdpq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-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>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> game. Therefore, it makes sense to define the <italic>critical bias</italic> of a game, <italic>b</italic><sup>*</sup>, to be the "breaking point" of the game. That is, Maker wins the (1, <italic>b</italic>) game whenever <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kf4x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>b</mml:mi><mml:mo><</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> and loses otherwise. An analogous definition of the critical bias exists for Avoider‐Enforcer games: here, the critical bias of a game <italic>b</italic><sup>*</sup> is such that Avoider wins the (1, <italic>b</italic>) game for every <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kf9h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-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:msup><mml:mi>b</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>, and loses otherwise.</p> <p>We prove that, for every <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kdzv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>ω</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>, </mml:mo><mml:mi>G</mml:mi><mml:mo>∼</mml:mo><mml:mi>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> is typically such that the critical bias for all the aforementioned Maker‐Breaker games is asymptotically <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kf3d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>. We also prove that in the case <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kfhm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>Θ</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>, the critical bias is <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kfnp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>Θ</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>. These results settle a conjecture of Stojaković and Szabó. For Avoider‐Enforcer games, we prove that for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kfb1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>, the critical bias for all the aforementioned games is <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7kqp2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20528:rsa20528-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>Θ</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46, 651–676, 2015</p> </abstract>