Randomly coloring planar graphs with fewer colors than the maximum degree1. Issue 4 (1st July 2014)

• Record Type: Journal Article
• Authors: Hayes, Thomas P.; Vera, Juan C.; Vigoda, Eric
• Is Part Of: Random structures & algorithms. Volume 47:Issue 4(2015)
• Page Start: 731
• Page End: 759
• Publication Date: 2014-07-01
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20560
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We study Markov chains for randomly sampling <italic>k</italic>‐colorings of a graph with maximum degree Δ. Our main result is a polynomial upper bound on the mixing time of the single‐site update chain known as the Glauber dynamics for planar graphs when <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk854" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20560:rsa20560-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo>Ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>Δ</mml:mo><mml:mo>/</mml:mo><mml:mi>log</mml:mi><mml:mo>Δ</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk833" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20560:rsa20560-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>Δ</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo>ϵ</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>, for fixed <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk8f8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20560:rsa20560-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ϵ</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>.</p> <p>The main challenge when <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk8dr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20560:rsa20560-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:mo>Δ</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> is the possibility of "frozen" vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk8c7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20560:rsa20560-math-0005" 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>, even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using "local uniformity" properties. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 731–759, 2015</p> </abstract>