{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:VKVCLID6KZUH3RM5JAI7IW7GBB","short_pith_number":"pith:VKVCLID6","canonical_record":{"source":{"id":"1901.01959","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-07T18:25:39Z","cross_cats_sorted":[],"title_canon_sha256":"df188a090f823b327b21043edea007036a8ad88126eb1dc273618c6e097f1486","abstract_canon_sha256":"200daac62f5dc5353cda0ff3d2595233369777b407ec4bca0c593be9a263648d"},"schema_version":"1.0"},"canonical_sha256":"aaaa25a07e56687dc59d4811f45be6084239d7f5fdec29165de11552e5751f61","source":{"kind":"arxiv","id":"1901.01959","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01959","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01959v1","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01959","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"pith_short_12","alias_value":"VKVCLID6KZUH","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VKVCLID6KZUH3RM5","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VKVCLID6","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:VKVCLID6KZUH3RM5JAI7IW7GBB","target":"record","payload":{"canonical_record":{"source":{"id":"1901.01959","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-07T18:25:39Z","cross_cats_sorted":[],"title_canon_sha256":"df188a090f823b327b21043edea007036a8ad88126eb1dc273618c6e097f1486","abstract_canon_sha256":"200daac62f5dc5353cda0ff3d2595233369777b407ec4bca0c593be9a263648d"},"schema_version":"1.0"},"canonical_sha256":"aaaa25a07e56687dc59d4811f45be6084239d7f5fdec29165de11552e5751f61","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:48.976160Z","signature_b64":"J95x6ua6WQOsd5nOuap6XP4qNkhC8aKYAnv9jpf5Am8pAMVT92s2BALo+HUrWB+z9qN9GPx/WO9J/0dZA5L5Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aaaa25a07e56687dc59d4811f45be6084239d7f5fdec29165de11552e5751f61","last_reissued_at":"2026-05-17T23:56:48.975743Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:48.975743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.01959","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:56:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OUI/QIgAt3UaSLuDTTTGzKETvyPBLhH4ZzVPrTzQ3vBheJ7TixjK/UhQkXnUK/KERn6XxgaQcuIZEZYh7MPHAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:26:30.431084Z"},"content_sha256":"c95b29a09e98498a1440a590c595f7e2271aebbc0e344cb739ecd00d587a6dd1","schema_version":"1.0","event_id":"sha256:c95b29a09e98498a1440a590c595f7e2271aebbc0e344cb739ecd00d587a6dd1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:VKVCLID6KZUH3RM5JAI7IW7GBB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Toughness and prism-hamiltonicity of $P_4$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"M. N. Ellingham, Pouria Salehi Nowbandegani, Songling Shan","submitted_at":"2019-01-07T18:25:39Z","abstract_excerpt":"The \\emph{prism} over a graph $G$ is the product $G \\Box K_2$, i.e., the graph obtained by taking two copies of $G$ and adding a perfect matching joining the two copies of each vertex by an edge. The graph $G$ is called \\emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every $1$-tough $P_4$-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for $k \\geq 1$ a $P_4$-free graph has a spanning \\emph{$k$-walk} (closed walk using each vertex at most $k$ times) if and only if it is $\\frac{1}{k}$-tough. As our main result, we sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01959","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:56:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3EYrUwdAcSCSgrF0c+2KZ9Z73OU2UumKUWoL8H+Ay2LmsTe+4aevEg/pwI2wRS0PX8vHjWpHe/xQqzrxzLtgAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:26:30.431796Z"},"content_sha256":"beb78e97eb80a839d84191856a1bc284fdd480d0db680079b5f1336f22965fdd","schema_version":"1.0","event_id":"sha256:beb78e97eb80a839d84191856a1bc284fdd480d0db680079b5f1336f22965fdd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VKVCLID6KZUH3RM5JAI7IW7GBB/bundle.json","state_url":"https://pith.science/pith/VKVCLID6KZUH3RM5JAI7IW7GBB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VKVCLID6KZUH3RM5JAI7IW7GBB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T18:26:30Z","links":{"resolver":"https://pith.science/pith/VKVCLID6KZUH3RM5JAI7IW7GBB","bundle":"https://pith.science/pith/VKVCLID6KZUH3RM5JAI7IW7GBB/bundle.json","state":"https://pith.science/pith/VKVCLID6KZUH3RM5JAI7IW7GBB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VKVCLID6KZUH3RM5JAI7IW7GBB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:VKVCLID6KZUH3RM5JAI7IW7GBB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"200daac62f5dc5353cda0ff3d2595233369777b407ec4bca0c593be9a263648d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-07T18:25:39Z","title_canon_sha256":"df188a090f823b327b21043edea007036a8ad88126eb1dc273618c6e097f1486"},"schema_version":"1.0","source":{"id":"1901.01959","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01959","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01959v1","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01959","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"pith_short_12","alias_value":"VKVCLID6KZUH","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VKVCLID6KZUH3RM5","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VKVCLID6","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:beb78e97eb80a839d84191856a1bc284fdd480d0db680079b5f1336f22965fdd","target":"graph","created_at":"2026-05-17T23:56:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The \\emph{prism} over a graph $G$ is the product $G \\Box K_2$, i.e., the graph obtained by taking two copies of $G$ and adding a perfect matching joining the two copies of each vertex by an edge. The graph $G$ is called \\emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every $1$-tough $P_4$-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for $k \\geq 1$ a $P_4$-free graph has a spanning \\emph{$k$-walk} (closed walk using each vertex at most $k$ times) if and only if it is $\\frac{1}{k}$-tough. As our main result, we sho","authors_text":"M. N. Ellingham, Pouria Salehi Nowbandegani, Songling Shan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-07T18:25:39Z","title":"Toughness and prism-hamiltonicity of $P_4$-free graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01959","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c95b29a09e98498a1440a590c595f7e2271aebbc0e344cb739ecd00d587a6dd1","target":"record","created_at":"2026-05-17T23:56:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"200daac62f5dc5353cda0ff3d2595233369777b407ec4bca0c593be9a263648d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-07T18:25:39Z","title_canon_sha256":"df188a090f823b327b21043edea007036a8ad88126eb1dc273618c6e097f1486"},"schema_version":"1.0","source":{"id":"1901.01959","kind":"arxiv","version":1}},"canonical_sha256":"aaaa25a07e56687dc59d4811f45be6084239d7f5fdec29165de11552e5751f61","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aaaa25a07e56687dc59d4811f45be6084239d7f5fdec29165de11552e5751f61","first_computed_at":"2026-05-17T23:56:48.975743Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:48.975743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J95x6ua6WQOsd5nOuap6XP4qNkhC8aKYAnv9jpf5Am8pAMVT92s2BALo+HUrWB+z9qN9GPx/WO9J/0dZA5L5Dg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:48.976160Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.01959","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c95b29a09e98498a1440a590c595f7e2271aebbc0e344cb739ecd00d587a6dd1","sha256:beb78e97eb80a839d84191856a1bc284fdd480d0db680079b5f1336f22965fdd"],"state_sha256":"8ecb4d0c1274a6e31b990d2bb7871e05d3d96a832ccd67300bde381abd69893f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xq0FZP/BdcofUCY1Q64J+JKyHtoc56gWRWG9VWOjM2a3D66P9izySdaJj3XnxlammKP+d5AHly4VwKMuqIADCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T18:26:30.436164Z","bundle_sha256":"e40c085bcf7e2f1373583f084401842c2edcbcac19bf0a7fa33423bbe62c46be"}}