{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:WRQPDCABHBSBPHOQNCSB7TIVVK","short_pith_number":"pith:WRQPDCAB","schema_version":"1.0","canonical_sha256":"b460f188013864179dd068a41fcd15aa98cb9a3c9abfaa019fa8cc046f00d5bb","source":{"kind":"arxiv","id":"2504.01501","version":2},"attestation_state":"computed","paper":{"title":"Vertex-Based Localization of Erd\\H{o}s-Gallai Theorems for Paths and Cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Bangalore), L. Sunil Chandran (Indian Institute of Science, Rajat Adak","submitted_at":"2025-04-02T08:52:28Z","abstract_excerpt":"For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \\leq \\frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \\in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \\[m \\leq \\sum_{v \\in V(G)} \\frac{p(v)}{2}\\] The Erd\\H{o}s-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the lo"},"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":"2504.01501","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-02T08:52:28Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"079d94712e00183ab0b1478b5bb3bf46df483a3b11df3f5e5a899159faaf438e","abstract_canon_sha256":"ce42bbd7209949ce44815d8a6371287fdf9a8f0fb41e54d79e1c024ec6d2d503"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:59:28.521890Z","signature_b64":"bcLi9EFzuVT3kBPeGbprCeBm3xGJSQ3rnXJSRIuYeT2BxnMhN+546WjrGuly63beLbbDKedjuWkjZeqD8+DABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b460f188013864179dd068a41fcd15aa98cb9a3c9abfaa019fa8cc046f00d5bb","last_reissued_at":"2026-07-05T10:59:28.521396Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:59:28.521396Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vertex-Based Localization of Erd\\H{o}s-Gallai Theorems for Paths and Cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Bangalore), L. Sunil Chandran (Indian Institute of Science, Rajat Adak","submitted_at":"2025-04-02T08:52:28Z","abstract_excerpt":"For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \\leq \\frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \\in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \\[m \\leq \\sum_{v \\in V(G)} \\frac{p(v)}{2}\\] The Erd\\H{o}s-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the lo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.01501","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.01501/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2504.01501","created_at":"2026-07-05T10:59:28.521455+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.01501v2","created_at":"2026-07-05T10:59:28.521455+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.01501","created_at":"2026-07-05T10:59:28.521455+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRQPDCABHBSB","created_at":"2026-07-05T10:59:28.521455+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRQPDCABHBSBPHOQ","created_at":"2026-07-05T10:59:28.521455+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRQPDCAB","created_at":"2026-07-05T10:59:28.521455+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.10467","citing_title":"An Upper Bound on the Linear Tur\\'{a}n Number of $k$-Crowns","ref_index":1,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK","json":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK.json","graph_json":"https://pith.science/api/pith-number/WRQPDCABHBSBPHOQNCSB7TIVVK/graph.json","events_json":"https://pith.science/api/pith-number/WRQPDCABHBSBPHOQNCSB7TIVVK/events.json","paper":"https://pith.science/paper/WRQPDCAB"},"agent_actions":{"view_html":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK","download_json":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK.json","view_paper":"https://pith.science/paper/WRQPDCAB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.01501&json=true","fetch_graph":"https://pith.science/api/pith-number/WRQPDCABHBSBPHOQNCSB7TIVVK/graph.json","fetch_events":"https://pith.science/api/pith-number/WRQPDCABHBSBPHOQNCSB7TIVVK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK/action/storage_attestation","attest_author":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK/action/author_attestation","sign_citation":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK/action/citation_signature","submit_replication":"https://pith.science/pith/WRQPDCABHBSBPHOQNCSB7TIVVK/action/replication_record"}},"created_at":"2026-07-05T10:59:28.521455+00:00","updated_at":"2026-07-05T10:59:28.521455+00:00"}