{"id":183183,"date":"2006-12-11T00:00:00","date_gmt":"2009-10-31T10:26:09","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/randomly-coloring-planar-graphs-with-fewer-colors-than-the-maximum-degree\/"},"modified":"2016-09-09T09:57:22","modified_gmt":"2016-09-09T16:57:22","slug":"randomly-coloring-planar-graphs-with-fewer-colors-than-the-maximum-degree","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/randomly-coloring-planar-graphs-with-fewer-colors-than-the-maximum-degree\/","title":{"rendered":"Randomly coloring planar graphs with fewer colors than the maximum degree"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We study Markov chains for randomly sampling k-colorings of a graph with maximum degree D.  We prove the first results, for a general class of graphs, showing fast convergence of such chains when the number of colors k << D.<\/p>\n<p>When D =O((log n)^(1+eps)) and k > D\/log D we prove O(n log n) mixing time of the single-site update chain, known as the Glauber dynamics, for any planar graph.  This result is tight, there exist planar graphs for which the Glauber dynamics has super-polynomial mixing time whenever k = o(D\/ log D).<\/p>\n<p>A challenging aspect of the case when D is constant and k  D\/ log D.  This implies polynomial mixing time of the original Glauber dynamics under the same conditions.<\/p>\n<p>Joint work with Thomas P. Hayes and Eric Vigoda.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We study Markov chains for randomly sampling k-colorings of a graph with maximum degree D. We prove the first results, for a general class of graphs, showing fast convergence of such chains when the number of colors k D\/log D we prove O(n log n) mixing time of the single-site update chain, known as the [&hellip;]<\/p>\n","protected":false},"featured_media":194938,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-183183","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/rPKcVvC61BI","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183183","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183183\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/194938"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=183183"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=183183"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=183183"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=183183"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=183183"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=183183"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=183183"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=183183"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=183183"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=183183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}