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The maximum average degree of a graph $G$, denoted by $\\mad(G)$, is the maximum of the average degree of all subgraphs of $G$. In this paper, it is proved that if $\\mad(G)<4$, then $\\chi'_a(G)\\leq{\\Delta(G)+2}$; if $\\mad(G)<3$, then $\\chi'_a(G)\\leq{\\Delta(G)+1}$. This implies that every triangle-free planar graph $G$ is acyclically edge $(\\Delta(G)+2)$-color"},"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":"1202.6129","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-28T05:54:52Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"c8a1a9a4eaa448891349a9bc01a60e91266ba2eb0222b461de673e372fb48988","abstract_canon_sha256":"7fc839493c9a5e75b881de2118c82e064b518974c838d69adaf28aa7f23666d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:11.126055Z","signature_b64":"lub69QZwKSzPgeuuc4CKoa3KH/vjDqAC/RjBv+viaKadDZGBoHdrsA3bTm3Ep6mqWOz0/2smca7iCfCThljdAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"723a120cdd6a4be3dc89591ca20efb209a43c50eca7049368cea777dcc2008c9","last_reissued_at":"2026-05-18T02:25:11.125394Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:11.125394Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Acyclic edge coloring of sparse graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jianfeng Hou","submitted_at":"2012-02-28T05:54:52Z","abstract_excerpt":"A proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. 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