{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:6UDO4HWAYHXIM5QHWJOIVKUG5X","short_pith_number":"pith:6UDO4HWA","canonical_record":{"source":{"id":"1601.04551","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-18T14:57:11Z","cross_cats_sorted":[],"title_canon_sha256":"12c5564342edb2bd9e19cc8a0c70dd14dade8442bccd449289fa47c9f803e94e","abstract_canon_sha256":"d46e371d92995359fcc19907d11f398b2d85b786a5274dc9f14e50a9933003c5"},"schema_version":"1.0"},"canonical_sha256":"f506ee1ec0c1ee867607b25c8aaa86eddaf84f31de43d4836a8acf1ac286fccb","source":{"kind":"arxiv","id":"1601.04551","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.04551","created_at":"2026-05-18T01:08:18Z"},{"alias_kind":"arxiv_version","alias_value":"1601.04551v2","created_at":"2026-05-18T01:08:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04551","created_at":"2026-05-18T01:08:18Z"},{"alias_kind":"pith_short_12","alias_value":"6UDO4HWAYHXI","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6UDO4HWAYHXIM5QH","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6UDO4HWA","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:6UDO4HWAYHXIM5QHWJOIVKUG5X","target":"record","payload":{"canonical_record":{"source":{"id":"1601.04551","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-18T14:57:11Z","cross_cats_sorted":[],"title_canon_sha256":"12c5564342edb2bd9e19cc8a0c70dd14dade8442bccd449289fa47c9f803e94e","abstract_canon_sha256":"d46e371d92995359fcc19907d11f398b2d85b786a5274dc9f14e50a9933003c5"},"schema_version":"1.0"},"canonical_sha256":"f506ee1ec0c1ee867607b25c8aaa86eddaf84f31de43d4836a8acf1ac286fccb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:18.621276Z","signature_b64":"8yvjnhVHSyYgZ3HlrfhhC5IxuyyFAP4Trr6HJh3fb35FsgqpenTQ3E57e1fCyaqGvStjtc4UcdBsaMNnRBmCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f506ee1ec0c1ee867607b25c8aaa86eddaf84f31de43d4836a8acf1ac286fccb","last_reissued_at":"2026-05-18T01:08:18.620828Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:18.620828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.04551","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":"2026-05-18T01:08:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GMyB2jvGd9D37ZDOkW7hBKZt4gs2LecIlYd5T5MHXHlKMy7DILGHZhaPJGtsPsYiCM9XBvf27MtB/EQBQUXIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T16:51:19.221928Z"},"content_sha256":"0e63d2454b4c7cba733eb7ce9bf517351cdf8b150012b7dca7605faa2d385e62","schema_version":"1.0","event_id":"sha256:0e63d2454b4c7cba733eb7ce9bf517351cdf8b150012b7dca7605faa2d385e62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:6UDO4HWAYHXIM5QHWJOIVKUG5X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Square-free graphs are multiplicative","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Marcin Wrochna","submitted_at":"2016-01-18T14:57:11Z","abstract_excerpt":"A graph K is square-free if it contains no four-cycle as a subgraph. A graph K is multiplicative if GxH -> K implies G -> K or H -> K, for all graphs G,H. Here GxH is the tensor (or categorical) graph product and G -> K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture states that all cliques K_n are multiplicative. However, the only non-trivial graphs known to be multiplicative are K_3, odd cycles, and still more generally, circular cliques $K_{p/q}$ with 2 <= p/q < 4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04551","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":"2026-05-18T01:08:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B9zpXk3MIYNBkavkXvcE4FFyBlg4dTBtxjKxGc0QPXP6GoBDS51OTd0aZefxZhbCR440XuX3QMumI0evt5brBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T16:51:19.222628Z"},"content_sha256":"0d9ea6be1fce2bc264fea69b099bb0471c75bf6e5fe57a97c280c626e80e8aba","schema_version":"1.0","event_id":"sha256:0d9ea6be1fce2bc264fea69b099bb0471c75bf6e5fe57a97c280c626e80e8aba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6UDO4HWAYHXIM5QHWJOIVKUG5X/bundle.json","state_url":"https://pith.science/pith/6UDO4HWAYHXIM5QHWJOIVKUG5X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6UDO4HWAYHXIM5QHWJOIVKUG5X/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":"2026-05-29T16:51:19Z","links":{"resolver":"https://pith.science/pith/6UDO4HWAYHXIM5QHWJOIVKUG5X","bundle":"https://pith.science/pith/6UDO4HWAYHXIM5QHWJOIVKUG5X/bundle.json","state":"https://pith.science/pith/6UDO4HWAYHXIM5QHWJOIVKUG5X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6UDO4HWAYHXIM5QHWJOIVKUG5X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:6UDO4HWAYHXIM5QHWJOIVKUG5X","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":"d46e371d92995359fcc19907d11f398b2d85b786a5274dc9f14e50a9933003c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-18T14:57:11Z","title_canon_sha256":"12c5564342edb2bd9e19cc8a0c70dd14dade8442bccd449289fa47c9f803e94e"},"schema_version":"1.0","source":{"id":"1601.04551","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.04551","created_at":"2026-05-18T01:08:18Z"},{"alias_kind":"arxiv_version","alias_value":"1601.04551v2","created_at":"2026-05-18T01:08:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04551","created_at":"2026-05-18T01:08:18Z"},{"alias_kind":"pith_short_12","alias_value":"6UDO4HWAYHXI","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6UDO4HWAYHXIM5QH","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6UDO4HWA","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:0d9ea6be1fce2bc264fea69b099bb0471c75bf6e5fe57a97c280c626e80e8aba","target":"graph","created_at":"2026-05-18T01:08:18Z","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":"A graph K is square-free if it contains no four-cycle as a subgraph. A graph K is multiplicative if GxH -> K implies G -> K or H -> K, for all graphs G,H. Here GxH is the tensor (or categorical) graph product and G -> K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture states that all cliques K_n are multiplicative. However, the only non-trivial graphs known to be multiplicative are K_3, odd cycles, and still more generally, circular cliques $K_{p/q}$ with 2 <= p/q < 4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In","authors_text":"Marcin Wrochna","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-18T14:57:11Z","title":"Square-free graphs are multiplicative"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04551","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":"sha256:0e63d2454b4c7cba733eb7ce9bf517351cdf8b150012b7dca7605faa2d385e62","target":"record","created_at":"2026-05-18T01:08:18Z","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":"d46e371d92995359fcc19907d11f398b2d85b786a5274dc9f14e50a9933003c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-18T14:57:11Z","title_canon_sha256":"12c5564342edb2bd9e19cc8a0c70dd14dade8442bccd449289fa47c9f803e94e"},"schema_version":"1.0","source":{"id":"1601.04551","kind":"arxiv","version":2}},"canonical_sha256":"f506ee1ec0c1ee867607b25c8aaa86eddaf84f31de43d4836a8acf1ac286fccb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f506ee1ec0c1ee867607b25c8aaa86eddaf84f31de43d4836a8acf1ac286fccb","first_computed_at":"2026-05-18T01:08:18.620828Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:18.620828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8yvjnhVHSyYgZ3HlrfhhC5IxuyyFAP4Trr6HJh3fb35FsgqpenTQ3E57e1fCyaqGvStjtc4UcdBsaMNnRBmCBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:18.621276Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.04551","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e63d2454b4c7cba733eb7ce9bf517351cdf8b150012b7dca7605faa2d385e62","sha256:0d9ea6be1fce2bc264fea69b099bb0471c75bf6e5fe57a97c280c626e80e8aba"],"state_sha256":"d91ef0a7d37b963cb70b996cce7788b5069d223f1b27c13f8c0582e33b34d677"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4rE1W4BWaJqhxz1SBX7+FikOSWBoyMivOsyKiKWXewPHDznnwjSjr3CiHFC+OWyWfJ/9zuEvbZOUCExDF+PcBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T16:51:19.226478Z","bundle_sha256":"ee655984c4ed7886dd4d8d9de1f9e5e8db80bc73266734748fd0fbfe17a01869"}}