{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:SXKFISD4R65AC66UC4L2ITPM7W","short_pith_number":"pith:SXKFISD4","canonical_record":{"source":{"id":"1505.00767","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-04T19:47:41Z","cross_cats_sorted":[],"title_canon_sha256":"d092ce9da1ce02c2005cea6075affb3127729563f596284d870bc09e78d348c5","abstract_canon_sha256":"078ca95155a31a53de84e52e0adf8645db3e2884fa35dd95dd1f42d3ef62435b"},"schema_version":"1.0"},"canonical_sha256":"95d454487c8fba017bd41717a44decfdb481826a0569396fc67e3d87d43914c2","source":{"kind":"arxiv","id":"1505.00767","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.00767","created_at":"2026-05-18T01:17:30Z"},{"alias_kind":"arxiv_version","alias_value":"1505.00767v2","created_at":"2026-05-18T01:17:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.00767","created_at":"2026-05-18T01:17:30Z"},{"alias_kind":"pith_short_12","alias_value":"SXKFISD4R65A","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SXKFISD4R65AC66U","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SXKFISD4","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:SXKFISD4R65AC66UC4L2ITPM7W","target":"record","payload":{"canonical_record":{"source":{"id":"1505.00767","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-04T19:47:41Z","cross_cats_sorted":[],"title_canon_sha256":"d092ce9da1ce02c2005cea6075affb3127729563f596284d870bc09e78d348c5","abstract_canon_sha256":"078ca95155a31a53de84e52e0adf8645db3e2884fa35dd95dd1f42d3ef62435b"},"schema_version":"1.0"},"canonical_sha256":"95d454487c8fba017bd41717a44decfdb481826a0569396fc67e3d87d43914c2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:30.768887Z","signature_b64":"XzTSCRHFr7+1pCSYmI1aLNJ1ylOB2QsKZUXXDe9HlBpFMHkdm9yzzfcrYS9hu1KzzDYXkkOgqt5BCeldvfCSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95d454487c8fba017bd41717a44decfdb481826a0569396fc67e3d87d43914c2","last_reissued_at":"2026-05-18T01:17:30.768118Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:30.768118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.00767","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tOWfOqaChixqI+DDxS3DkYywvPmoouN+03Zwbh6AHQ4IqOdUxgy2BGTZTnM1mFcv39IQFI6UHMAKz6LT8/PrDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:50:59.942515Z"},"content_sha256":"ab0f04b96502f029155a59837410011044ecebcd918402d1d2fdb1043ecbacf0","schema_version":"1.0","event_id":"sha256:ab0f04b96502f029155a59837410011044ecebcd918402d1d2fdb1043ecbacf0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:SXKFISD4R65AC66UC4L2ITPM7W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graphs with many strong orientations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Paul Horn, Sinan Aksoy","submitted_at":"2015-05-04T19:47:41Z","abstract_excerpt":"We establish mild conditions under which a possibly irregular, sparse graph $G$ has \"many\" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)\\log_2{n}$ and has Cheeger constant at least $c_2\\frac{\\log_2\\log_2{n}}{\\log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00767","kind":"arxiv","version":2},"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":"2026-05-18T01:17:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HW+b+4Tbu2X1jk0meTg+iKdwyEScNgOQbo86XzabO8d5IDh8TM144nL7DYJzHriURJtGYNIVVBbSDC7AOKb5Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:50:59.942867Z"},"content_sha256":"53b57c2003b4bab5f61bc5cdc42f3bbbd2f643cfcc1ca71362185ee0a109f9dd","schema_version":"1.0","event_id":"sha256:53b57c2003b4bab5f61bc5cdc42f3bbbd2f643cfcc1ca71362185ee0a109f9dd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SXKFISD4R65AC66UC4L2ITPM7W/bundle.json","state_url":"https://pith.science/pith/SXKFISD4R65AC66UC4L2ITPM7W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SXKFISD4R65AC66UC4L2ITPM7W/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":"2026-06-03T07:50:59Z","links":{"resolver":"https://pith.science/pith/SXKFISD4R65AC66UC4L2ITPM7W","bundle":"https://pith.science/pith/SXKFISD4R65AC66UC4L2ITPM7W/bundle.json","state":"https://pith.science/pith/SXKFISD4R65AC66UC4L2ITPM7W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SXKFISD4R65AC66UC4L2ITPM7W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:SXKFISD4R65AC66UC4L2ITPM7W","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":"078ca95155a31a53de84e52e0adf8645db3e2884fa35dd95dd1f42d3ef62435b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-04T19:47:41Z","title_canon_sha256":"d092ce9da1ce02c2005cea6075affb3127729563f596284d870bc09e78d348c5"},"schema_version":"1.0","source":{"id":"1505.00767","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.00767","created_at":"2026-05-18T01:17:30Z"},{"alias_kind":"arxiv_version","alias_value":"1505.00767v2","created_at":"2026-05-18T01:17:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.00767","created_at":"2026-05-18T01:17:30Z"},{"alias_kind":"pith_short_12","alias_value":"SXKFISD4R65A","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SXKFISD4R65AC66U","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SXKFISD4","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:53b57c2003b4bab5f61bc5cdc42f3bbbd2f643cfcc1ca71362185ee0a109f9dd","target":"graph","created_at":"2026-05-18T01:17:30Z","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":"We establish mild conditions under which a possibly irregular, sparse graph $G$ has \"many\" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)\\log_2{n}$ and has Cheeger constant at least $c_2\\frac{\\log_2\\log_2{n}}{\\log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we prov","authors_text":"Paul Horn, Sinan Aksoy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-04T19:47:41Z","title":"Graphs with many strong orientations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00767","kind":"arxiv","version":2},"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:ab0f04b96502f029155a59837410011044ecebcd918402d1d2fdb1043ecbacf0","target":"record","created_at":"2026-05-18T01:17:30Z","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":"078ca95155a31a53de84e52e0adf8645db3e2884fa35dd95dd1f42d3ef62435b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-04T19:47:41Z","title_canon_sha256":"d092ce9da1ce02c2005cea6075affb3127729563f596284d870bc09e78d348c5"},"schema_version":"1.0","source":{"id":"1505.00767","kind":"arxiv","version":2}},"canonical_sha256":"95d454487c8fba017bd41717a44decfdb481826a0569396fc67e3d87d43914c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95d454487c8fba017bd41717a44decfdb481826a0569396fc67e3d87d43914c2","first_computed_at":"2026-05-18T01:17:30.768118Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:30.768118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XzTSCRHFr7+1pCSYmI1aLNJ1ylOB2QsKZUXXDe9HlBpFMHkdm9yzzfcrYS9hu1KzzDYXkkOgqt5BCeldvfCSBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:30.768887Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.00767","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ab0f04b96502f029155a59837410011044ecebcd918402d1d2fdb1043ecbacf0","sha256:53b57c2003b4bab5f61bc5cdc42f3bbbd2f643cfcc1ca71362185ee0a109f9dd"],"state_sha256":"57d09ac0eed4b309d145e6110710905f2c03179e62b96119de255b72bbf47db0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J+w2KeCQw0WrmXWr2n+hWSOrZJDbrxo+Ze2xK8T4Pcid8VCAOurskbASzEUPWX+f88HLA1zHjbWj2CDMfRhODQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T07:50:59.944832Z","bundle_sha256":"ec7a400108a13bd0e11ad57baa4867fb58d96ddc44db0ae9c676170f2ccafe03"}}