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Moreover we prove that:\n  $a'(G)\\le \\lceil 6.42(\\Delta-1)\\rceil$ if $G$ has girth $g\\ge 5\\,$; $a'(G)\\le \\lceil5.77 (\\Delta-1)\\rc$ if\n  $G$ has girth $g\\ge 7$; $a'(G)\\le \\lc4.52(\\D-1)\\rc$ if $g\\ge 53$;\n  $a'(G)\\le \\D+2\\,$ if $g\\ge \\lceil25.84\\D\\log\\D(1+ 4.1/\\log\\D)\\rceil$.\n  We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that\n  $a(G)\\le \\lc 6.59 \\Delta^{4/3}+3.3\\D\\rc$. We also prove that the st"},"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":"1005.1875","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-11T16:05:46Z","cross_cats_sorted":[],"title_canon_sha256":"caaa94b66ac90050cb49af246357e4f7d70bb0da52d520c7c1a1bbb2c7f1af59","abstract_canon_sha256":"48b74eded966e7b8421a0af5aa829f5c333d737d73a6740fb4bea2942cc6d519"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:56.897714Z","signature_b64":"S6xQL4iJnsazva1tOjxcMFpLKnasmIXHZ2zOczHin1CqH67Z2EF+TRpwL5c350cfJX1DYvaXm6M1JPIPfLIyCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b231bb2210587e27f526a12cf7a285529fe0bf65fdb557b1591a396de6db690a","last_reissued_at":"2026-05-18T04:06:56.896998Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:56.896998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved bounds on coloring of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aldo Procacci, Benedetto Scoppola, Sokol Ndreca","submitted_at":"2010-05-11T16:05:46Z","abstract_excerpt":"Given a graph $G$ with maximum degree $\\Delta\\ge 3$, we prove that the acyclic edge chromatic number $a'(G)$ of $G$ is such that $a'(G)\\le\\lceil 9.62 (\\Delta-1)\\rceil$. Moreover we prove that:\n  $a'(G)\\le \\lceil 6.42(\\Delta-1)\\rceil$ if $G$ has girth $g\\ge 5\\,$; $a'(G)\\le \\lceil5.77 (\\Delta-1)\\rc$ if\n  $G$ has girth $g\\ge 7$; $a'(G)\\le \\lc4.52(\\D-1)\\rc$ if $g\\ge 53$;\n  $a'(G)\\le \\D+2\\,$ if $g\\ge \\lceil25.84\\D\\log\\D(1+ 4.1/\\log\\D)\\rceil$.\n  We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that\n  $a(G)\\le \\lc 6.59 \\Delta^{4/3}+3.3\\D\\rc$. We also prove that the st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1875","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":"1005.1875","created_at":"2026-05-18T04:06:56.897123+00:00"},{"alias_kind":"arxiv_version","alias_value":"1005.1875v2","created_at":"2026-05-18T04:06:56.897123+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.1875","created_at":"2026-05-18T04:06:56.897123+00:00"},{"alias_kind":"pith_short_12","alias_value":"WIY3WIQQLB7C","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"WIY3WIQQLB7CP5JG","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"WIY3WIQQ","created_at":"2026-05-18T12:26:15.391820+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/WIY3WIQQLB7CP5JGUEWPPIUFKK","json":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK.json","graph_json":"https://pith.science/api/pith-number/WIY3WIQQLB7CP5JGUEWPPIUFKK/graph.json","events_json":"https://pith.science/api/pith-number/WIY3WIQQLB7CP5JGUEWPPIUFKK/events.json","paper":"https://pith.science/paper/WIY3WIQQ"},"agent_actions":{"view_html":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK","download_json":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK.json","view_paper":"https://pith.science/paper/WIY3WIQQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1005.1875&json=true","fetch_graph":"https://pith.science/api/pith-number/WIY3WIQQLB7CP5JGUEWPPIUFKK/graph.json","fetch_events":"https://pith.science/api/pith-number/WIY3WIQQLB7CP5JGUEWPPIUFKK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK/action/storage_attestation","attest_author":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK/action/author_attestation","sign_citation":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK/action/citation_signature","submit_replication":"https://pith.science/pith/WIY3WIQQLB7CP5JGUEWPPIUFKK/action/replication_record"}},"created_at":"2026-05-18T04:06:56.897123+00:00","updated_at":"2026-05-18T04:06:56.897123+00:00"}