{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:WP6YA3OP3Q5LXMA2NABTDIYVZE","short_pith_number":"pith:WP6YA3OP","canonical_record":{"source":{"id":"1211.1467","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-07T06:54:27Z","cross_cats_sorted":["cs.IT","math.CO","math.IT"],"title_canon_sha256":"0c0d9683af00b18996d2ca3fd27412e95d52fd8e39d39e03cdaa27c872c97127","abstract_canon_sha256":"1d038163a62d26d010c310f4ca0f903eecd062877858e27f7abe98024f2b9036"},"schema_version":"1.0"},"canonical_sha256":"b3fd806dcfdc3abbb01a680331a315c915d8113ef59825346961ec80f20a3451","source":{"kind":"arxiv","id":"1211.1467","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1467","created_at":"2026-05-18T03:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1467v4","created_at":"2026-05-18T03:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1467","created_at":"2026-05-18T03:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"WP6YA3OP3Q5L","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WP6YA3OP3Q5LXMA2","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WP6YA3OP","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:WP6YA3OP3Q5LXMA2NABTDIYVZE","target":"record","payload":{"canonical_record":{"source":{"id":"1211.1467","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-07T06:54:27Z","cross_cats_sorted":["cs.IT","math.CO","math.IT"],"title_canon_sha256":"0c0d9683af00b18996d2ca3fd27412e95d52fd8e39d39e03cdaa27c872c97127","abstract_canon_sha256":"1d038163a62d26d010c310f4ca0f903eecd062877858e27f7abe98024f2b9036"},"schema_version":"1.0"},"canonical_sha256":"b3fd806dcfdc3abbb01a680331a315c915d8113ef59825346961ec80f20a3451","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:14:17.472622Z","signature_b64":"/VK7ppdqdvvAjGP1dGLSRVmnfF7xkmXkQzirvmxEuHLTkrseV+FlfJj3NWQc5CSesN8QzHYUUJbXy9T6fUgCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3fd806dcfdc3abbb01a680331a315c915d8113ef59825346961ec80f20a3451","last_reissued_at":"2026-05-18T03:14:17.471926Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:14:17.471926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.1467","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C992b2kLb9CCJvwfTrvC/u0eTxxCcud1j34AK2xOgnXY9etE3WYZpyuPT+KdLX7++ww8MVMLoA2/WxkM2fNcBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:02:02.267300Z"},"content_sha256":"fe6549ef31a36d530924092948d25c8023cddb0c08420b7e2dd715b8db7da20e","schema_version":"1.0","event_id":"sha256:fe6549ef31a36d530924092948d25c8023cddb0c08420b7e2dd715b8db7da20e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:WP6YA3OP3Q5LXMA2NABTDIYVZE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Edge distribution in generalized graph products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.CO","math.IT"],"primary_cat":"cs.DM","authors_text":"Dan Vilenchik, Michael Langberg","submitted_at":"2012-11-07T06:54:27Z","abstract_excerpt":"Given a graph $G=(V,E)$, an integer $k$, and a function $f_G:V^k \\times V^k \\to {0,1}$, the $k^{th}$ graph product of $G$ w.r.t $f_G$ is the graph with vertex set $V^k$, and an edge between two vertices $x=(x_1,...,x_k)$ and $y=(y_1,...,y_k)$ iff $f_G(x,y)=1$. Graph products are a basic combinatorial object, widely studied and used in different areas such as hardness of approximation, information theory, etc. We study graph products for functions $f_G$ of the form $f_G(x,y)=1$ iff there are at least $t$ indices $i \\in [k]$ s.t. $(x_i,y_i)\\in E$, where $t \\in [k]$ is a fixed parameter in $f_G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1467","kind":"arxiv","version":4},"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":"2026-05-18T03:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3u+imCxLQen09MsNYv9DbTpOlFQM7pgqa5bGk2bak8mmv2Vs9AnPh5dXWUhjev4uY5XwBJ2/IjRklNLDciEUAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:02:02.267671Z"},"content_sha256":"8a51282203b11480139e2b66f60fa90f31bcb82f5496d1570085516fc312f2c6","schema_version":"1.0","event_id":"sha256:8a51282203b11480139e2b66f60fa90f31bcb82f5496d1570085516fc312f2c6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/bundle.json","state_url":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/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":"2026-06-01T17:02:02Z","links":{"resolver":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE","bundle":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/bundle.json","state":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:WP6YA3OP3Q5LXMA2NABTDIYVZE","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":"1d038163a62d26d010c310f4ca0f903eecd062877858e27f7abe98024f2b9036","cross_cats_sorted":["cs.IT","math.CO","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-07T06:54:27Z","title_canon_sha256":"0c0d9683af00b18996d2ca3fd27412e95d52fd8e39d39e03cdaa27c872c97127"},"schema_version":"1.0","source":{"id":"1211.1467","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1467","created_at":"2026-05-18T03:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1467v4","created_at":"2026-05-18T03:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1467","created_at":"2026-05-18T03:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"WP6YA3OP3Q5L","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WP6YA3OP3Q5LXMA2","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WP6YA3OP","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:8a51282203b11480139e2b66f60fa90f31bcb82f5496d1570085516fc312f2c6","target":"graph","created_at":"2026-05-18T03:14:17Z","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":"Given a graph $G=(V,E)$, an integer $k$, and a function $f_G:V^k \\times V^k \\to {0,1}$, the $k^{th}$ graph product of $G$ w.r.t $f_G$ is the graph with vertex set $V^k$, and an edge between two vertices $x=(x_1,...,x_k)$ and $y=(y_1,...,y_k)$ iff $f_G(x,y)=1$. Graph products are a basic combinatorial object, widely studied and used in different areas such as hardness of approximation, information theory, etc. We study graph products for functions $f_G$ of the form $f_G(x,y)=1$ iff there are at least $t$ indices $i \\in [k]$ s.t. $(x_i,y_i)\\in E$, where $t \\in [k]$ is a fixed parameter in $f_G$.","authors_text":"Dan Vilenchik, Michael Langberg","cross_cats":["cs.IT","math.CO","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-07T06:54:27Z","title":"Edge distribution in generalized graph products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1467","kind":"arxiv","version":4},"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:fe6549ef31a36d530924092948d25c8023cddb0c08420b7e2dd715b8db7da20e","target":"record","created_at":"2026-05-18T03:14:17Z","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":"1d038163a62d26d010c310f4ca0f903eecd062877858e27f7abe98024f2b9036","cross_cats_sorted":["cs.IT","math.CO","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-07T06:54:27Z","title_canon_sha256":"0c0d9683af00b18996d2ca3fd27412e95d52fd8e39d39e03cdaa27c872c97127"},"schema_version":"1.0","source":{"id":"1211.1467","kind":"arxiv","version":4}},"canonical_sha256":"b3fd806dcfdc3abbb01a680331a315c915d8113ef59825346961ec80f20a3451","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b3fd806dcfdc3abbb01a680331a315c915d8113ef59825346961ec80f20a3451","first_computed_at":"2026-05-18T03:14:17.471926Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:14:17.471926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/VK7ppdqdvvAjGP1dGLSRVmnfF7xkmXkQzirvmxEuHLTkrseV+FlfJj3NWQc5CSesN8QzHYUUJbXy9T6fUgCCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:14:17.472622Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.1467","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe6549ef31a36d530924092948d25c8023cddb0c08420b7e2dd715b8db7da20e","sha256:8a51282203b11480139e2b66f60fa90f31bcb82f5496d1570085516fc312f2c6"],"state_sha256":"c718d487c05d80699c2c4c4009c47fe9c921ff4d22d2b3d2e7cc5cf5d3dbc224"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7An7XEp2lTUX5loTWSBAshLalqgrPsRUlQ/TdB2H0y/IFqs3waQ8JczoJjDVkEUmXvclT90BNyJLIvT2agSFCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T17:02:02.269654Z","bundle_sha256":"995f579294f5358003eb6ac1ec86e4d83f9422ca0453737c62972c3d2afb7219"}}