{"paper":{"title":"Randomly coloring planar graphs with fewer colors than the maximum degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Eric Vigoda, Juan C. Vera, Thomas P. Hayes","submitted_at":"2007-06-11T17:55:42Z","abstract_excerpt":"We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $\\Delta$. 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 $k=\\Omega(\\Delta/\\log{\\Delta})$. 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 $\\Delta^{1-\\eps}$, for fixed $\\eps > 0$.\n  The main challenge when $k \\le \\Delta + 1$ is the possibility of \"frozen\" vertices, that is, vertices for which only one color is possib"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0706.1530","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}