{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:EJ6K56ESEHKUZGXNHGKNAR4TBM","short_pith_number":"pith:EJ6K56ES","canonical_record":{"source":{"id":"2407.01198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-01T11:47:34Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"f724ec539a616ef2522af4bb15b9f26c218264fbaad6477e45956eafb9e83edd","abstract_canon_sha256":"d4d6a2d0c129d51bb50aa933ec4ab03a853e11d5a21057e7ade6e3a67386c159"},"schema_version":"1.0"},"canonical_sha256":"227caef89221d54c9aed3994d047930b2697a5e839baabad923d894ce4fdfd21","source":{"kind":"arxiv","id":"2407.01198","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01198","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01198v1","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01198","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"pith_short_12","alias_value":"EJ6K56ESEHKU","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"pith_short_16","alias_value":"EJ6K56ESEHKUZGXN","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"pith_short_8","alias_value":"EJ6K56ES","created_at":"2026-07-05T08:38:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:EJ6K56ESEHKUZGXNHGKNAR4TBM","target":"record","payload":{"canonical_record":{"source":{"id":"2407.01198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-01T11:47:34Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"f724ec539a616ef2522af4bb15b9f26c218264fbaad6477e45956eafb9e83edd","abstract_canon_sha256":"d4d6a2d0c129d51bb50aa933ec4ab03a853e11d5a21057e7ade6e3a67386c159"},"schema_version":"1.0"},"canonical_sha256":"227caef89221d54c9aed3994d047930b2697a5e839baabad923d894ce4fdfd21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:38:44.478705Z","signature_b64":"IkT0HvPoqJaipDy7mbk++zZqkCbPX9WL3ZTQI6SbgA+nf2ztz1B0ddz1MoBRGmzN6bNdzAxBwdAQV6w/goK1Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"227caef89221d54c9aed3994d047930b2697a5e839baabad923d894ce4fdfd21","last_reissued_at":"2026-07-05T08:38:44.478271Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:38:44.478271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.01198","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":"2026-07-05T08:38:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DJlPN/GlHy9KaSHGBDe1rA6mi0qXES+pNoTEcYWXBPH2Of07GXhPdSfZyDid5KhJIx2X4ZIIa28L9eAbFv45BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T05:41:05.780280Z"},"content_sha256":"4873d1757336054bc0f4dd064c06bf83e1ad1fcdf62bc579b267de98e6f29398","schema_version":"1.0","event_id":"sha256:4873d1757336054bc0f4dd064c06bf83e1ad1fcdf62bc579b267de98e6f29398"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:EJ6K56ESEHKUZGXNHGKNAR4TBM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cycles of weight divisible by $k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Ajit A. Diwan","submitted_at":"2024-07-01T11:47:34Z","abstract_excerpt":"A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order $f(k)$ of a weighted complete directed graph that does not contain a directed cycle of weight divisible by $k$, for an integer $k \\ge 2$, was raised by Alon and Krivelevich [J. Graph Theory 98 (2021) 623-629]. They showed that $f(k)$ is $O(k\\log k)$ and $f(k) \\le 2k-2$ if $k$ is prime. The best bounds known to us are $f(k) \\le 2k-2$ for all $k$ and $f(k) < (3k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01198","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.01198/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:38:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xikou9FYSucYN8qedUpTC0BjFVNy/9N6Wg63lpzjvaAM6J03ZmCoD/tA2z/SH1+ViTxqixVtPtcaKeS+fj5+Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T05:41:05.781058Z"},"content_sha256":"c713efadc23bfaca7c18b02e9b0845c9e115b63320691c77eb2cd581f27c4000","schema_version":"1.0","event_id":"sha256:c713efadc23bfaca7c18b02e9b0845c9e115b63320691c77eb2cd581f27c4000"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EJ6K56ESEHKUZGXNHGKNAR4TBM/bundle.json","state_url":"https://pith.science/pith/EJ6K56ESEHKUZGXNHGKNAR4TBM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EJ6K56ESEHKUZGXNHGKNAR4TBM/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":"2026-08-20T05:41:05Z","links":{"resolver":"https://pith.science/pith/EJ6K56ESEHKUZGXNHGKNAR4TBM","bundle":"https://pith.science/pith/EJ6K56ESEHKUZGXNHGKNAR4TBM/bundle.json","state":"https://pith.science/pith/EJ6K56ESEHKUZGXNHGKNAR4TBM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EJ6K56ESEHKUZGXNHGKNAR4TBM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:EJ6K56ESEHKUZGXNHGKNAR4TBM","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":"d4d6a2d0c129d51bb50aa933ec4ab03a853e11d5a21057e7ade6e3a67386c159","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-01T11:47:34Z","title_canon_sha256":"f724ec539a616ef2522af4bb15b9f26c218264fbaad6477e45956eafb9e83edd"},"schema_version":"1.0","source":{"id":"2407.01198","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01198","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01198v1","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01198","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"pith_short_12","alias_value":"EJ6K56ESEHKU","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"pith_short_16","alias_value":"EJ6K56ESEHKUZGXN","created_at":"2026-07-05T08:38:44Z"},{"alias_kind":"pith_short_8","alias_value":"EJ6K56ES","created_at":"2026-07-05T08:38:44Z"}],"graph_snapshots":[{"event_id":"sha256:c713efadc23bfaca7c18b02e9b0845c9e115b63320691c77eb2cd581f27c4000","target":"graph","created_at":"2026-07-05T08:38:44Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2407.01198/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order $f(k)$ of a weighted complete directed graph that does not contain a directed cycle of weight divisible by $k$, for an integer $k \\ge 2$, was raised by Alon and Krivelevich [J. Graph Theory 98 (2021) 623-629]. They showed that $f(k)$ is $O(k\\log k)$ and $f(k) \\le 2k-2$ if $k$ is prime. The best bounds known to us are $f(k) \\le 2k-2$ for all $k$ and $f(k) < (3k","authors_text":"Ajit A. Diwan","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-01T11:47:34Z","title":"Cycles of weight divisible by $k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01198","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":"sha256:4873d1757336054bc0f4dd064c06bf83e1ad1fcdf62bc579b267de98e6f29398","target":"record","created_at":"2026-07-05T08:38:44Z","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":"d4d6a2d0c129d51bb50aa933ec4ab03a853e11d5a21057e7ade6e3a67386c159","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-01T11:47:34Z","title_canon_sha256":"f724ec539a616ef2522af4bb15b9f26c218264fbaad6477e45956eafb9e83edd"},"schema_version":"1.0","source":{"id":"2407.01198","kind":"arxiv","version":1}},"canonical_sha256":"227caef89221d54c9aed3994d047930b2697a5e839baabad923d894ce4fdfd21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"227caef89221d54c9aed3994d047930b2697a5e839baabad923d894ce4fdfd21","first_computed_at":"2026-07-05T08:38:44.478271Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:38:44.478271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IkT0HvPoqJaipDy7mbk++zZqkCbPX9WL3ZTQI6SbgA+nf2ztz1B0ddz1MoBRGmzN6bNdzAxBwdAQV6w/goK1Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T08:38:44.478705Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.01198","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4873d1757336054bc0f4dd064c06bf83e1ad1fcdf62bc579b267de98e6f29398","sha256:c713efadc23bfaca7c18b02e9b0845c9e115b63320691c77eb2cd581f27c4000"],"state_sha256":"46a7cd4876d84bfca4290a35312a276161496f2473c32afeef128f3f9d1e9b7e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v2GtVYN3wCE9Lm2L9ACfMT+6tTukCgWYkRXRn85EpGFV88l8LPV4dXTY4Rk7dyTzosC1kCj+e5CvDDYGcrPNBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T05:41:05.789407Z","bundle_sha256":"7054ca814589d1835455d6b5ad5d0fc210bee1723f78694df700feee368a23e4"}}