{"paper":{"title":"Constraining the clustering transition for colorings of sparse random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Frieze, Michael Anastos, Wesley Pegden","submitted_at":"2017-05-22T18:44:35Z","abstract_excerpt":"Let $\\Omega_q$ denote the set of proper $q$-colorings of the random graph $G_{n,m}, m=dn/2$ and let $H_q$ be the graph with vertex set $\\Omega_q$ and an edge $\\{\\sigma,\\tau\\}$ where $\\sigma,\\tau$ are mappings $[n]\\to[q]$ iff $h(\\sigma,\\tau)=1$. Here $h(\\sigma,\\tau)$ is the Hamming distance $|\\{v\\in [n]:\\sigma(v)\\neq\\tau(v)\\}|$. We show that w.h.p. $H_q$ contains a single giant component containing almost all colorings in $\\Omega_q$ if $d$ is sufficiently large and $q\\geq \\frac{cd}{\\log d}$ for a constant $c>3/2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.07944","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"}