{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:CZQZPC2QLP2GYEFXE7JZ2WZKOI","short_pith_number":"pith:CZQZPC2Q","schema_version":"1.0","canonical_sha256":"1661978b505bf46c10b727d39d5b2a7231c063eafe02469097fdea952ce468e5","source":{"kind":"arxiv","id":"1610.03476","version":1},"attestation_state":"computed","paper":{"title":"On the number of cycles in a graph with restricted cycle lengths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Keszegh, Bal\\'azs Patk\\'os, Cory Palmer, D\\'aniel Gerbner","submitted_at":"2016-10-11T19:48:27Z","abstract_excerpt":"Let $L$ be a set of positive integers. We call a (directed) graph $G$ an $L$\\emph{-cycle graph} if all cycle lengths in $G$ belong to $L$. Let $c(L,n)$ be the maximum number of cycles possible in an $n$-vertex $L$-cycle graph (we use $\\vec{c}(L,n)$ for the number of cycles in directed graphs). In the undirected case we show that for any fixed set $L$, we have $c(L,n)=\\Theta_L(n^{\\lfloor k/\\ell \\rfloor})$ where $k$ is the largest element of $L$ and $2\\ell$ is the smallest even element of $L$ (if $L$ contains only odd elements, then $c(L,n)=\\Theta_L(n)$ holds.) We also give a characterization of"},"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":"1610.03476","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-11T19:48:27Z","cross_cats_sorted":[],"title_canon_sha256":"28d5d632c5128b7f6c9bc81c5f8fbfe6d5e5742399f2ea115e2af5cb2b6f78aa","abstract_canon_sha256":"64bb3d5c039841937720edf4476f272282f3a919431200b1c41d80bd22b0ff2c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:31.357024Z","signature_b64":"EgwcBY6EvzJonPDkUrf7Je0sorWvY9RaOZ0LbFN4fUfbIQQY+sQu3zjsK1LXrzWUeuJQGkvu21JVx9H0ozyvCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1661978b505bf46c10b727d39d5b2a7231c063eafe02469097fdea952ce468e5","last_reissued_at":"2026-05-18T01:02:31.356469Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:31.356469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of cycles in a graph with restricted cycle lengths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Keszegh, Bal\\'azs Patk\\'os, Cory Palmer, D\\'aniel Gerbner","submitted_at":"2016-10-11T19:48:27Z","abstract_excerpt":"Let $L$ be a set of positive integers. We call a (directed) graph $G$ an $L$\\emph{-cycle graph} if all cycle lengths in $G$ belong to $L$. Let $c(L,n)$ be the maximum number of cycles possible in an $n$-vertex $L$-cycle graph (we use $\\vec{c}(L,n)$ for the number of cycles in directed graphs). In the undirected case we show that for any fixed set $L$, we have $c(L,n)=\\Theta_L(n^{\\lfloor k/\\ell \\rfloor})$ where $k$ is the largest element of $L$ and $2\\ell$ is the smallest even element of $L$ (if $L$ contains only odd elements, then $c(L,n)=\\Theta_L(n)$ holds.) We also give a characterization of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03476","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1610.03476","created_at":"2026-05-18T01:02:31.356567+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.03476v1","created_at":"2026-05-18T01:02:31.356567+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.03476","created_at":"2026-05-18T01:02:31.356567+00:00"},{"alias_kind":"pith_short_12","alias_value":"CZQZPC2QLP2G","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"CZQZPC2QLP2GYEFX","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"CZQZPC2Q","created_at":"2026-05-18T12:30:09.641336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI","json":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI.json","graph_json":"https://pith.science/api/pith-number/CZQZPC2QLP2GYEFXE7JZ2WZKOI/graph.json","events_json":"https://pith.science/api/pith-number/CZQZPC2QLP2GYEFXE7JZ2WZKOI/events.json","paper":"https://pith.science/paper/CZQZPC2Q"},"agent_actions":{"view_html":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI","download_json":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI.json","view_paper":"https://pith.science/paper/CZQZPC2Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.03476&json=true","fetch_graph":"https://pith.science/api/pith-number/CZQZPC2QLP2GYEFXE7JZ2WZKOI/graph.json","fetch_events":"https://pith.science/api/pith-number/CZQZPC2QLP2GYEFXE7JZ2WZKOI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI/action/storage_attestation","attest_author":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI/action/author_attestation","sign_citation":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI/action/citation_signature","submit_replication":"https://pith.science/pith/CZQZPC2QLP2GYEFXE7JZ2WZKOI/action/replication_record"}},"created_at":"2026-05-18T01:02:31.356567+00:00","updated_at":"2026-05-18T01:02:31.356567+00:00"}