{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:USK3LMIXPDAWHZMWW5PJX52SQI","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":"82916701448ff2e9ddb80c290491d43347dfdf2e50bbc90ff72da2cf3203665f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-27T06:46:13Z","title_canon_sha256":"151fc2bd58a901321b590bec5641b779faabdcaf39154deab90f14da911522ba"},"schema_version":"1.0","source":{"id":"1902.10357","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.10357","created_at":"2026-05-17T23:52:31Z"},{"alias_kind":"arxiv_version","alias_value":"1902.10357v1","created_at":"2026-05-17T23:52:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.10357","created_at":"2026-05-17T23:52:31Z"},{"alias_kind":"pith_short_12","alias_value":"USK3LMIXPDAW","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"USK3LMIXPDAWHZMW","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"USK3LMIX","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:88cae44d9d691a82bc9f985e67b6946a91e7e22c9c861cfc0c049d5f0e8de813","target":"graph","created_at":"2026-05-17T23:52:31Z","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":"The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the Sunlet graph, denoted $\\mathcal{S}_n$, and the Star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of $\\mathcal{S}_n \\Box K_{1,2}$ is $n$, and the crossing number of $\\mathcal{S}_n \\Box K_{1,3}$ is $3n$. An upper bound for the crossing number of $\\mathcal{S}_n \\Box K_{1,m}$ is also given.","authors_text":"Alex Newcombe, Michael Haythorpe","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-27T06:46:13Z","title":"On the Crossing Number of the Cartesian Product of a Sunlet Graph and a Star Graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.10357","kind":"arxiv","version":1},"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:c060a9f4ad90d4d78c7ce566bf2e929b130ae34e70ea0b6a0c64d86307b0adfe","target":"record","created_at":"2026-05-17T23:52:31Z","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":"82916701448ff2e9ddb80c290491d43347dfdf2e50bbc90ff72da2cf3203665f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-27T06:46:13Z","title_canon_sha256":"151fc2bd58a901321b590bec5641b779faabdcaf39154deab90f14da911522ba"},"schema_version":"1.0","source":{"id":"1902.10357","kind":"arxiv","version":1}},"canonical_sha256":"a495b5b11778c163e596b75e9bf7528225b1461382736f2474630e71a250d8f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a495b5b11778c163e596b75e9bf7528225b1461382736f2474630e71a250d8f1","first_computed_at":"2026-05-17T23:52:31.070641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:31.070641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dxxL58e+MUMPlyEKV77KqHww4JzrG7YQncNC9RRHk8Fw6W4eZqIrE3uDSawMl45sCyVCw2+EECur6SdnQNbxDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:31.071050Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.10357","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c060a9f4ad90d4d78c7ce566bf2e929b130ae34e70ea0b6a0c64d86307b0adfe","sha256:88cae44d9d691a82bc9f985e67b6946a91e7e22c9c861cfc0c049d5f0e8de813"],"state_sha256":"ade6f6e2933f16fea143db7f4615917661616a4806c7d93bbfb46b9803460ee7"}