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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 $f_G$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1211.1467","created_at":"2026-05-18T03:14:17.472010+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.1467v4","created_at":"2026-05-18T03:14:17.472010+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1467","created_at":"2026-05-18T03:14:17.472010+00:00"},{"alias_kind":"pith_short_12","alias_value":"WP6YA3OP3Q5L","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"WP6YA3OP3Q5LXMA2","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"WP6YA3OP","created_at":"2026-05-18T12:27:25.539911+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE","json":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE.json","graph_json":"https://pith.science/api/pith-number/WP6YA3OP3Q5LXMA2NABTDIYVZE/graph.json","events_json":"https://pith.science/api/pith-number/WP6YA3OP3Q5LXMA2NABTDIYVZE/events.json","paper":"https://pith.science/paper/WP6YA3OP"},"agent_actions":{"view_html":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE","download_json":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE.json","view_paper":"https://pith.science/paper/WP6YA3OP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.1467&json=true","fetch_graph":"https://pith.science/api/pith-number/WP6YA3OP3Q5LXMA2NABTDIYVZE/graph.json","fetch_events":"https://pith.science/api/pith-number/WP6YA3OP3Q5LXMA2NABTDIYVZE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/action/storage_attestation","attest_author":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/action/author_attestation","sign_citation":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/action/citation_signature","submit_replication":"https://pith.science/pith/WP6YA3OP3Q5LXMA2NABTDIYVZE/action/replication_record"}},"created_at":"2026-05-18T03:14:17.472010+00:00","updated_at":"2026-05-18T03:14:17.472010+00:00"}