{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:6AISCAIG3Y6RVEN47ELVPJOBNY","short_pith_number":"pith:6AISCAIG","canonical_record":{"source":{"id":"1806.06315","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-16T23:58:47Z","cross_cats_sorted":[],"title_canon_sha256":"9055a992ee103271d2c7d41ff2294827d043f6d1af45acec1c78014cdebec7de","abstract_canon_sha256":"c61054293b261ee433dd3345f6aa8d80e5ddebfdac8f9a855e20e505297eb3cc"},"schema_version":"1.0"},"canonical_sha256":"f011210106de3d1a91bcf91757a5c16e096c3a9a377f1e777e968a6cfd0d43c6","source":{"kind":"arxiv","id":"1806.06315","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.06315","created_at":"2026-05-18T00:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1806.06315v5","created_at":"2026-05-18T00:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06315","created_at":"2026-05-18T00:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"6AISCAIG3Y6R","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6AISCAIG3Y6RVEN4","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6AISCAIG","created_at":"2026-05-18T12:32:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:6AISCAIG3Y6RVEN47ELVPJOBNY","target":"record","payload":{"canonical_record":{"source":{"id":"1806.06315","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-16T23:58:47Z","cross_cats_sorted":[],"title_canon_sha256":"9055a992ee103271d2c7d41ff2294827d043f6d1af45acec1c78014cdebec7de","abstract_canon_sha256":"c61054293b261ee433dd3345f6aa8d80e5ddebfdac8f9a855e20e505297eb3cc"},"schema_version":"1.0"},"canonical_sha256":"f011210106de3d1a91bcf91757a5c16e096c3a9a377f1e777e968a6cfd0d43c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:31.855079Z","signature_b64":"3n2V/N5aKrve3aTTlztB1sGdz955Km0vivoFKST3+eE2bw8YW6cWI367YNchMSDgiYhUBzu78YUfpWug39dICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f011210106de3d1a91bcf91757a5c16e096c3a9a377f1e777e968a6cfd0d43c6","last_reissued_at":"2026-05-18T00:03:31.854448Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:31.854448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.06315","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TRRgDtUm2HNjrBWMLnkFQST2gWGd8x8REaR0RB3QwDHW6VrDDTY7X20B0aUhPPkHEdIcRPVn+wdVbOLlFj8mBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T16:19:58.039514Z"},"content_sha256":"6c3f1a9e6cbb31087bf4edc9c2a93c81508921dad12577d373a71572bd62ee9d","schema_version":"1.0","event_id":"sha256:6c3f1a9e6cbb31087bf4edc9c2a93c81508921dad12577d373a71572bd62ee9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:6AISCAIG3Y6RVEN47ELVPJOBNY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"ON $(\\triangle, 1)$-GRAPHS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Kelmans, Rafael Aparicio","submitted_at":"2018-06-16T23:58:47Z","abstract_excerpt":"Let $G = (V, E)$ be a graph and $\\lambda $ a non-negative integer. A graph $G$ is called a $(\\lambda, 1)$-{\\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\\lambda$ triangles, and $(c2)$ every two non-adjacent vertices in $G$ are the end-vertices of exactly one two-edge path in $G$. It turns out that there are infinitely many feasible 4-tuples $(v, d, \\lambda, 1)$ with $\\lambda \\ge 1$. On the other hand (and this is our main result), there is no $(v, d, \\lambda, 1)$-graphs with $\\lambda \\ge 1$. As a byproduct, we obtain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06315","kind":"arxiv","version":5},"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":"2026-05-18T00:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cBL7rq9DCUynODaPIpMMVajFWHI6eczBp3sIibmTHvgZZtu8S3CN5GRy4vFwkOTUgz+1X17sEC4ETCrHmGWbDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T16:19:58.039856Z"},"content_sha256":"78149daf119cbffa33c9d2a8cb5702ef52cb987318ca30202d37756388287e51","schema_version":"1.0","event_id":"sha256:78149daf119cbffa33c9d2a8cb5702ef52cb987318ca30202d37756388287e51"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/bundle.json","state_url":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/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":"2026-07-02T16:19:58Z","links":{"resolver":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY","bundle":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/bundle.json","state":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6AISCAIG3Y6RVEN47ELVPJOBNY","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":"c61054293b261ee433dd3345f6aa8d80e5ddebfdac8f9a855e20e505297eb3cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-16T23:58:47Z","title_canon_sha256":"9055a992ee103271d2c7d41ff2294827d043f6d1af45acec1c78014cdebec7de"},"schema_version":"1.0","source":{"id":"1806.06315","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.06315","created_at":"2026-05-18T00:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1806.06315v5","created_at":"2026-05-18T00:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06315","created_at":"2026-05-18T00:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"6AISCAIG3Y6R","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6AISCAIG3Y6RVEN4","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6AISCAIG","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:78149daf119cbffa33c9d2a8cb5702ef52cb987318ca30202d37756388287e51","target":"graph","created_at":"2026-05-18T00:03:31Z","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":"Let $G = (V, E)$ be a graph and $\\lambda $ a non-negative integer. A graph $G$ is called a $(\\lambda, 1)$-{\\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\\lambda$ triangles, and $(c2)$ every two non-adjacent vertices in $G$ are the end-vertices of exactly one two-edge path in $G$. It turns out that there are infinitely many feasible 4-tuples $(v, d, \\lambda, 1)$ with $\\lambda \\ge 1$. On the other hand (and this is our main result), there is no $(v, d, \\lambda, 1)$-graphs with $\\lambda \\ge 1$. As a byproduct, we obtain","authors_text":"Alexander Kelmans, Rafael Aparicio","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-16T23:58:47Z","title":"ON $(\\triangle, 1)$-GRAPHS"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06315","kind":"arxiv","version":5},"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":"sha256:6c3f1a9e6cbb31087bf4edc9c2a93c81508921dad12577d373a71572bd62ee9d","target":"record","created_at":"2026-05-18T00:03:31Z","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":"c61054293b261ee433dd3345f6aa8d80e5ddebfdac8f9a855e20e505297eb3cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-16T23:58:47Z","title_canon_sha256":"9055a992ee103271d2c7d41ff2294827d043f6d1af45acec1c78014cdebec7de"},"schema_version":"1.0","source":{"id":"1806.06315","kind":"arxiv","version":5}},"canonical_sha256":"f011210106de3d1a91bcf91757a5c16e096c3a9a377f1e777e968a6cfd0d43c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f011210106de3d1a91bcf91757a5c16e096c3a9a377f1e777e968a6cfd0d43c6","first_computed_at":"2026-05-18T00:03:31.854448Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:31.854448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3n2V/N5aKrve3aTTlztB1sGdz955Km0vivoFKST3+eE2bw8YW6cWI367YNchMSDgiYhUBzu78YUfpWug39dICw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:31.855079Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.06315","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c3f1a9e6cbb31087bf4edc9c2a93c81508921dad12577d373a71572bd62ee9d","sha256:78149daf119cbffa33c9d2a8cb5702ef52cb987318ca30202d37756388287e51"],"state_sha256":"13ed34c7058704e5be1f22cbf99a129716b4bfc41a972247c0ce5023ee57e074"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n08V3uY4+apOGbH2PxyeqsQLNf1BA/ay3ZXrIM77PEQntD/n+eZkSHlMie5tPKyEA+3kMqp7oexgJm0A+rpvAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T16:19:58.041730Z","bundle_sha256":"1328aa260be6dc0ac16ea30aef1ba3d23960d85aee5c5c13da299c6909eba412"}}