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In this work, we show that Gallai's conjecture holds in $\\mathcal{G}_{k}$, for every $k \\geq 3$. Further, we prove that for every graph $G$ in $\\mathcal{G}_{k}$ on $n$ vertices, there exists a partition of its edge set into $n/2$ paths of lengths in $\\{2k-1,2k,2k+1\\}$."},"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":"1510.02526","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-10-08T23:10:00Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"8b5e6165c5d2b577af353c20c86aec2f462d150066d7eded29fd1f1aecdf58cc","abstract_canon_sha256":"ef20a2d0fd9916dbfff67f15acb2ef331df038a3a9c730fd93d6976fca3d307e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:42.191968Z","signature_b64":"GKh2KdR3krgrK88ZVZVcEFgitJO3yz/QBpsj6M4xvab4tBnaSA7qmt/kIiJZ2MG1QHgwmdzHEWEQCBCblW4EBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc8344098e04d9307db2c73ab7690a261c4abf157a8492048760163d8400f9e7","last_reissued_at":"2026-05-18T01:30:42.191327Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:42.191327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On path decompositions of 2k-regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Andrea Jim\\'enez, F\\'abio Botler","submitted_at":"2015-10-08T23:10:00Z","abstract_excerpt":"Tibor Gallai conjectured that the edge set of every connected graph $G$ on $n$ vertices can be partitioned into $\\lceil n/2\\rceil$ paths. 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