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Here we study the ratio $r(G_{pt})=|P(G_{pt},\\tau+1)|/(\\tau-1)^{n-5}$ for a variety of planar triangulations. We construct infinite recursive families of planar triangulations $G_{pt,m}$ depending on a parameter $m$ linearly related to $n$ and show that if $P(G_{pt,m},q)$ only involves a single power of a polynomial, then $r(G_{pt,m})$ approaches zero exponentially fast "},"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":"1110.5883","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-10-26T19:14:42Z","cross_cats_sorted":["math.CO","math.MP"],"title_canon_sha256":"935a29103c96580158f36889d51e3655aa71206ecbc253a9f105daf6fb431940","abstract_canon_sha256":"3328bac9d45ccac00fbb6ddbfe34f822edefd56f861c0ae16e182e676810b1c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:15.030818Z","signature_b64":"kFUUkuCDY0M+Q2BFQcawThc3Dg+tImzrnARzMhUEkbgWxzl1AvRUF67wbVSb07sGKqVDMSQKxVpQUNeRf6irBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe8ae7b4c33141fd67cfe715da345b8b6ec403a1cac20a99a29ede14385c58e9","last_reissued_at":"2026-05-18T04:04:15.030038Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:15.030038Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chromatic Polynomials of Planar Triangulations, the Tutte Upper Bound, and Chromatic Zeros","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MP"],"primary_cat":"math-ph","authors_text":"Robert Shrock, Yan Xu","submitted_at":"2011-10-26T19:14:42Z","abstract_excerpt":"Tutte proved that if $G_{pt}$ is a planar triangulation and $P(G_{pt},q)$ is its chromatic polynomial, then $|P(G_{pt},\\tau+1)| \\le (\\tau-1)^{n-5}$, where $\\tau=(1+\\sqrt{5} \\,)/2$ and $n$ is the number of vertices in $G_{pt}$. Here we study the ratio $r(G_{pt})=|P(G_{pt},\\tau+1)|/(\\tau-1)^{n-5}$ for a variety of planar triangulations. We construct infinite recursive families of planar triangulations $G_{pt,m}$ depending on a parameter $m$ linearly related to $n$ and show that if $P(G_{pt,m},q)$ only involves a single power of a polynomial, then $r(G_{pt,m})$ approaches zero exponentially fast "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.5883","kind":"arxiv","version":1},"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":"1110.5883","created_at":"2026-05-18T04:04:15.030161+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.5883v1","created_at":"2026-05-18T04:04:15.030161+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.5883","created_at":"2026-05-18T04:04:15.030161+00:00"},{"alias_kind":"pith_short_12","alias_value":"72FOPNGDGFA7","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"72FOPNGDGFA72Z6P","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"72FOPNGD","created_at":"2026-05-18T12:26:22.705136+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/72FOPNGDGFA72Z6P44K5UNC3RN","json":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN.json","graph_json":"https://pith.science/api/pith-number/72FOPNGDGFA72Z6P44K5UNC3RN/graph.json","events_json":"https://pith.science/api/pith-number/72FOPNGDGFA72Z6P44K5UNC3RN/events.json","paper":"https://pith.science/paper/72FOPNGD"},"agent_actions":{"view_html":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN","download_json":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN.json","view_paper":"https://pith.science/paper/72FOPNGD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.5883&json=true","fetch_graph":"https://pith.science/api/pith-number/72FOPNGDGFA72Z6P44K5UNC3RN/graph.json","fetch_events":"https://pith.science/api/pith-number/72FOPNGDGFA72Z6P44K5UNC3RN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN/action/storage_attestation","attest_author":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN/action/author_attestation","sign_citation":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN/action/citation_signature","submit_replication":"https://pith.science/pith/72FOPNGDGFA72Z6P44K5UNC3RN/action/replication_record"}},"created_at":"2026-05-18T04:04:15.030161+00:00","updated_at":"2026-05-18T04:04:15.030161+00:00"}