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We improve here this bound by showing that \\[ \\pi(G,x) \\leq (x)_{\\downarrow k} (x-1)^{\\Delta(G)-k+1} x^{n-1-\\Delta(G)}\\] for every $x\\in \\mathbb{N},$ where $\\Delta(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\\geq 4$ then $\\pi(G,x)$ is at most $(x)_{\\downarrow k}(x-1)^{n-k}$ for every real $x\\geq n-2+\\left( {n \\"},"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":"1611.09545","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-29T10:05:57Z","cross_cats_sorted":[],"title_canon_sha256":"d3644a70d63c34f397645c9f070f713ebc93602c39e318f4a1817dca68293d19","abstract_canon_sha256":"9df116f8c06c0bf39302ee54b65e80cc75f22e66d5b5286536abc3c5af305ee2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:18.130347Z","signature_b64":"qRfvdg5n45/JIRlD44kCfltOqNzvnotAGuNNuPZY6pBlO4g0tUSOtx4OvrAAzCHWs+bH9U5j+xwN1PEA+q36BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f95bd8013942485cabf82429879eaa11e20a44a18102912e1d8605de1595cdec","last_reissued_at":"2026-05-18T00:56:18.129672Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:18.129672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Bounds for Chromatic Polynomials and Chromatic Roots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aysel Erey, Jason Brown","submitted_at":"2016-11-29T10:05:57Z","abstract_excerpt":"If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $\\pi(G,x)$, is at most $x(x-1)\\cdots (x-(k-1))x^{n-k} = (x)_{\\downarrow k}x^{n-k}$ for every $x\\in \\mathbb{N}$. We improve here this bound by showing that \\[ \\pi(G,x) \\leq (x)_{\\downarrow k} (x-1)^{\\Delta(G)-k+1} x^{n-1-\\Delta(G)}\\] for every $x\\in \\mathbb{N},$ where $\\Delta(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\\geq 4$ then $\\pi(G,x)$ is at most $(x)_{\\downarrow k}(x-1)^{n-k}$ for every real $x\\geq n-2+\\left( {n \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.09545","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":"1611.09545","created_at":"2026-05-18T00:56:18.129779+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.09545v1","created_at":"2026-05-18T00:56:18.129779+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.09545","created_at":"2026-05-18T00:56:18.129779+00:00"},{"alias_kind":"pith_short_12","alias_value":"7FN5QAJZIJEF","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"7FN5QAJZIJEFZK7Y","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"7FN5QAJZ","created_at":"2026-05-18T12:30:04.600751+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/7FN5QAJZIJEFZK7YEQUYPHVKCH","json":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH.json","graph_json":"https://pith.science/api/pith-number/7FN5QAJZIJEFZK7YEQUYPHVKCH/graph.json","events_json":"https://pith.science/api/pith-number/7FN5QAJZIJEFZK7YEQUYPHVKCH/events.json","paper":"https://pith.science/paper/7FN5QAJZ"},"agent_actions":{"view_html":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH","download_json":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH.json","view_paper":"https://pith.science/paper/7FN5QAJZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.09545&json=true","fetch_graph":"https://pith.science/api/pith-number/7FN5QAJZIJEFZK7YEQUYPHVKCH/graph.json","fetch_events":"https://pith.science/api/pith-number/7FN5QAJZIJEFZK7YEQUYPHVKCH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH/action/storage_attestation","attest_author":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH/action/author_attestation","sign_citation":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH/action/citation_signature","submit_replication":"https://pith.science/pith/7FN5QAJZIJEFZK7YEQUYPHVKCH/action/replication_record"}},"created_at":"2026-05-18T00:56:18.129779+00:00","updated_at":"2026-05-18T00:56:18.129779+00:00"}