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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 $c>3/2$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1705.07944","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-22T18:44:35Z","cross_cats_sorted":[],"title_canon_sha256":"66af8069362795e62fd6f91ecb51473e633d682560d926bc80bca1e4b4f8cecd","abstract_canon_sha256":"99b68e2ef9a96ed33f40019477360f74bc75d43c669e24171395b7cdd26b9030"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:21.357828Z","signature_b64":"vWEB3r6WO/HYZr9E/zSHB6s+ga0LnR9vMZdBti5sn52DXaib8y5mUTMIhOMdF/93ymN9GtkVJ6ExvquRQzg/BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cfc60dbfdb9a493403e0ef415a10af5ad2901f3833b30aa773ce804a5739451","last_reissued_at":"2026-05-18T00:26:21.357192Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:21.357192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1705.07944","created_at":"2026-05-18T00:26:21.357290+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.07944v2","created_at":"2026-05-18T00:26:21.357290+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.07944","created_at":"2026-05-18T00:26:21.357290+00:00"},{"alias_kind":"pith_short_12","alias_value":"TT6GBW75XGSJ","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_16","alias_value":"TT6GBW75XGSJGQB6","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_8","alias_value":"TT6GBW75","created_at":"2026-05-18T12:31:46.661854+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W","json":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W.json","graph_json":"https://pith.science/api/pith-number/TT6GBW75XGSJGQB6B32BLIIK6W/graph.json","events_json":"https://pith.science/api/pith-number/TT6GBW75XGSJGQB6B32BLIIK6W/events.json","paper":"https://pith.science/paper/TT6GBW75"},"agent_actions":{"view_html":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W","download_json":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W.json","view_paper":"https://pith.science/paper/TT6GBW75","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.07944&json=true","fetch_graph":"https://pith.science/api/pith-number/TT6GBW75XGSJGQB6B32BLIIK6W/graph.json","fetch_events":"https://pith.science/api/pith-number/TT6GBW75XGSJGQB6B32BLIIK6W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W/action/storage_attestation","attest_author":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W/action/author_attestation","sign_citation":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W/action/citation_signature","submit_replication":"https://pith.science/pith/TT6GBW75XGSJGQB6B32BLIIK6W/action/replication_record"}},"created_at":"2026-05-18T00:26:21.357290+00:00","updated_at":"2026-05-18T00:26:21.357290+00:00"}