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A graph G is said to have the F_k property if for any set of k distinct vertices {x_1, x_2, ..., x_k} in G, there is a hamiltonian path from x_1 to x_2 in G^2 containing k-2 distinct edges of G of the form x_iz_i, i = 3, ..., k. It was proved many years ago that every 2-connected graph has the F_3 property. In the first part of this work, we extend this result by proving that every 2-connected DT-graph has the F_4 property (Theorem 2) and will show"},"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":"1706.04414","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-14T11:26:11Z","cross_cats_sorted":[],"title_canon_sha256":"aa2ab410dc25af059a7944d54bbd2f179ca57481f311c5cc558f5b8c8371eb32","abstract_canon_sha256":"fedc13eb84ceedd6180f8a73ea1c3d07a09937a4aa971f9ccc74af2cc6889185"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:22.731513Z","signature_b64":"NdwyeOGsXCyzDdDWdF14XDYX++tProG7tFvKaMjlt9DIKGyH5G4a2lonTs9Fr5jLtuCt0R0hWPqpI6al0oZrDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab3913c5135a324aad15b243f921a6f9882d7afae9f76ea0b1208beccbc77495","last_reissued_at":"2026-05-18T00:42:22.730872Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:22.730872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Revisiting the Hamiltonian Theme in the Square of a Block: The Case of DT-Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gek L. Chia, Herbert Fleischner, Jan Ekstein","submitted_at":"2017-06-14T11:26:11Z","abstract_excerpt":"The square of a graph G, denoted G^2, is the graph obtained from G by joining by an edge any two nonadjacent vertices which have a common neighbor. A graph G is said to have the F_k property if for any set of k distinct vertices {x_1, x_2, ..., x_k} in G, there is a hamiltonian path from x_1 to x_2 in G^2 containing k-2 distinct edges of G of the form x_iz_i, i = 3, ..., k. It was proved many years ago that every 2-connected graph has the F_3 property. 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