{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:HH6UEIQPSNNW7SWIYPTLDXNJNX","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":"101cb878473afa42f52a17ab2a7b95151ec387895203a44c8f26fe8f357a93fc","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T12:11:45Z","title_canon_sha256":"dd5d2ef55b1aa46a94b1d1bcd545ca8a620435043081fde9aefca82b65404d55"},"schema_version":"1.0","source":{"id":"2605.21077","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21077","created_at":"2026-05-21T01:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21077v1","created_at":"2026-05-21T01:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21077","created_at":"2026-05-21T01:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"HH6UEIQPSNNW","created_at":"2026-05-21T01:05:35Z"},{"alias_kind":"pith_short_16","alias_value":"HH6UEIQPSNNW7SWI","created_at":"2026-05-21T01:05:35Z"},{"alias_kind":"pith_short_8","alias_value":"HH6UEIQP","created_at":"2026-05-21T01:05:35Z"}],"graph_snapshots":[{"event_id":"sha256:3dd07e9404df6aa4eb406e91fd785e702bc0cf25a8d6d27b4e099e45a3375f28","target":"graph","created_at":"2026-05-21T01:05:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.21077/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Galluccio--Loebl and Tesler showed that the perfect-matching polynomial of a graph embedded in an orientable surface of genus $g$ can be written as a linear combination of at most $4^g$ Pfaffians. We show that, in general, exponentially many Pfaffians are necessary. More precisely, among all graphs of orientable genus at most $g$, the maximum possible Pfaffian number is at least $(8/3)^g$. This lower bound holds even for connected matching-covered graphs. We also obtain exponential lower bounds for the Pfaffian number of complete bipartite graphs, and hence for even complete graphs, improving ","authors_text":"Priyanshu Pant, Ranveer Singh","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T12:11:45Z","title":"Exponential Lower Bounds for the Pfaffian Number of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21077","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":"sha256:6cd1b1a5e558e26c82abd833858d5a27a2bf448d62e0d01b12969e7e272140c8","target":"record","created_at":"2026-05-21T01:05:35Z","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":"101cb878473afa42f52a17ab2a7b95151ec387895203a44c8f26fe8f357a93fc","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T12:11:45Z","title_canon_sha256":"dd5d2ef55b1aa46a94b1d1bcd545ca8a620435043081fde9aefca82b65404d55"},"schema_version":"1.0","source":{"id":"2605.21077","kind":"arxiv","version":1}},"canonical_sha256":"39fd42220f935b6fcac8c3e6b1dda96dd1d68a4e6e6c52d958300b2889485746","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"39fd42220f935b6fcac8c3e6b1dda96dd1d68a4e6e6c52d958300b2889485746","first_computed_at":"2026-05-21T01:05:35.501202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:05:35.501202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z7/IJz33TtgUXN4gBoWYhLinK6BmoZTA0WCS7jOsGLs7Fv4VpVUSXTz1p5XzcW7XbFbm47yk9pTk/HKq65n7Ag==","signature_status":"signed_v1","signed_at":"2026-05-21T01:05:35.501926Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.21077","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6cd1b1a5e558e26c82abd833858d5a27a2bf448d62e0d01b12969e7e272140c8","sha256:3dd07e9404df6aa4eb406e91fd785e702bc0cf25a8d6d27b4e099e45a3375f28"],"state_sha256":"223d57d4333311efa4db9875023f31e428598af2631ad2bcebf0aedaf885684c"}