{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:HL3ZN4HZWVPZRBAAYNIOIHDAHV","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":"da1046cb4572db88bd3b915932b3715ccdd057b40eaaeabfd88473a5321600bf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-01-20T20:14:15Z","title_canon_sha256":"120a0c9ca1cfe19af35004785bee3bae26bc30157571e4d4e41b3147b5b80eb2"},"schema_version":"1.0","source":{"id":"1001.3661","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.3661","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"arxiv_version","alias_value":"1001.3661v2","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.3661","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"pith_short_12","alias_value":"HL3ZN4HZWVPZ","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"HL3ZN4HZWVPZRBAA","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"HL3ZN4HZ","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:abeda319f8355857556604edf0e0e99b57ab65837b713388cbf9eaca10d60101","target":"graph","created_at":"2026-05-18T03:41:40Z","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":"In this paper we study a new product of graphs called {\\em tight product}. A graph $H$ is said to be a tight product of two (undirected multi) graphs $G_1$ and $G_2$, if $V(H)=V(G_1)\\times V(G_2)$ and both projection maps $V(H)\\to V(G_1)$ and $V(H)\\to V(G_2)$ are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is $NP$-hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class-1 $(2k+1)$-regular graphs. We also obtain a new model of random $d$-regular graphs whose second eig","authors_text":"Amit Daniely, Nathan Linial","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-01-20T20:14:15Z","title":"Tight products and Expansion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.3661","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":"sha256:3a1a2489ef9d3b4ffc480454aed2f86879a40f7870c968b52ff640e05981c6c3","target":"record","created_at":"2026-05-18T03:41:40Z","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":"da1046cb4572db88bd3b915932b3715ccdd057b40eaaeabfd88473a5321600bf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-01-20T20:14:15Z","title_canon_sha256":"120a0c9ca1cfe19af35004785bee3bae26bc30157571e4d4e41b3147b5b80eb2"},"schema_version":"1.0","source":{"id":"1001.3661","kind":"arxiv","version":2}},"canonical_sha256":"3af796f0f9b55f988400c350e41c603d5f5dc8dd815488b6f58efb2a17efa244","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3af796f0f9b55f988400c350e41c603d5f5dc8dd815488b6f58efb2a17efa244","first_computed_at":"2026-05-18T03:41:40.586244Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:40.586244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+YcYaPBZl/Qyv9pVn480liFypiH0YNO2EaMfaxQ5D8Hz1HfPzDfY2AKGfq8HaSK0VB1P9fUv6/PL8Sa9bS+LDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:40.587162Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.3661","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3a1a2489ef9d3b4ffc480454aed2f86879a40f7870c968b52ff640e05981c6c3","sha256:abeda319f8355857556604edf0e0e99b57ab65837b713388cbf9eaca10d60101"],"state_sha256":"60ec9c9166a8fe6a52fe3d361f5137a1ebdf69ffbb252c2edfa7da28a8e0da57"}