Coloring graphs from random lists of fixed size. Issue 3 (24th September 2012)

• Record Type: Journal Article
• Authors: Casselgren, Carl Johan
• Is Part Of: Random structures & algorithms. Volume 44:Issue 3(2014)
• Page Start: 317
• Page End: 327
• Publication Date: 2012-09-24
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20469
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Let <italic>G</italic> = <italic>G</italic>(<italic>n</italic>) be a graph on <italic>n</italic> vertices with maximum degree bounded by some absolute constant Δ. Assign to each vertex <italic>v</italic> of <italic>G</italic> a list <italic>L</italic>(<italic>v</italic>) of colors by choosing each list uniformly at random from all <italic>k</italic>‐subsets of a color set <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1cjn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">C</mml:mi></mml:math></alternatives> of size <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1cmr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>σ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>. Such a list assignment is called a <italic>random</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1cf0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi mathvariant="script">C</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives><italic>‐list assignment</italic>. In this paper, we are interested in determining the asymptotic probability (as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1ch3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives>) of the existence of a proper coloring ϕ of <italic>G</italic>, such that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1c87" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>φ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∈</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> for every vertex <italic>v</italic> of <italic>G</italic>. We show, for all fixed <italic>k</italic> and growing <italic>n</italic>, that if <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1ccw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>σ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>, then the probability that <italic>G</italic> has such a proper coloring tends to 1 as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1d2f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives>. A similar result for complete graphs is also obtained: if <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1d30" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>σ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>1.223</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives> and <italic>L</italic> is a random <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1cxq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>, </mml:mo><mml:mi mathvariant="script">C</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>‐list assignment for the complete graph <italic>K</italic><sub><italic>n</italic></sub> on <italic>n</italic> vertices, then the probability that <italic>K</italic><sub><italic>n</italic></sub> has a proper coloring with colors from the random lists tends to 1 as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s1g1x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20469:rsa20469-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives>.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 317‐327, 2014</p> </abstract>