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Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G)\\le k+\\frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture was previously proved for $d=k+1$ and for $k=1$ when $d \\le 6$. We prove it for all $d$ when $k \\le 2$, except for $(k,d)=(2,1)$."},"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":"1502.04755","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-17T00:19:38Z","cross_cats_sorted":[],"title_canon_sha256":"f429f179d000837d326b5962bd6cafff97ce49b54bd9e5f177afc29cb6a40a72","abstract_canon_sha256":"e539c1da6a6e17bbabd32799a277c8bd42ad8715baac70eca6772ca475dbdc20"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:58.406979Z","signature_b64":"wT4ENdclACUgVkmreblGMBvJPRd08A0We1T44fbLkNj8zLIaxED1aEK7WqKkTpZsPO1nSns62wE+fR4pMrOjDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e58a9294466a4a8c540686cb81ad24179e404a7791d95fd66ab7832a7afe3fb","last_reissued_at":"2026-05-18T02:26:58.406589Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:58.406589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposition of Sparse Graphs into Forests: The Nine Dragon Tree Conjecture for $k \\le 2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Douglas B. 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