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We prove that every triangle-free planar graph admits an $({\\cal F},{\\cal F}_5)$-partition. Moreover we show that if for some integer $d$ there exists a triangle-free planar graph that does not admit an $({\\cal F},{\\cal F}_d)$-partition, then it is an NP-complete problem to decide whether a triangle-free planar graph admits such a partition."},"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":"1601.01523","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-01-07T12:58:25Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"80561f59c84caf3b14a83b512b98bec5e1d33668055810f91eec17d10bcfe4ff","abstract_canon_sha256":"4fa08382f61d5d7e981be56394772e515a252310fb13f96b5f7b9732382e6f37"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:13.397773Z","signature_b64":"Po6keLqZCR+PwKT3yxsfORuh5GaEFYLoAU44xGiavXHHAh5tzMbjJ5pVbh67hw6MfNOf8iHoLpPEbQkglDkhDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"749132ed8449863e195e4783bb16f82013b9ccf15fbc93d6bc3462d3f4753983","last_reissued_at":"2026-05-18T01:23:13.397130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:13.397130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partitioning a triangle-free planar graph into a forest and a forest of bounded degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Alexandre Pinlou, Fran\\c{c}ois Dross, Mickael Montassier","submitted_at":"2016-01-07T12:58:25Z","abstract_excerpt":"An $({\\cal F},{\\cal F}_d)$-partition of a graph is a vertex-partition into two sets $F$ and $F_d$ such that the graph induced by $F$ is a forest and the one induced by $F_d$ is a forest with maximum degree at most $d$. We prove that every triangle-free planar graph admits an $({\\cal F},{\\cal F}_5)$-partition. 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