{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:7F7SR3DNKVZRPV64YNCZILLXWG","short_pith_number":"pith:7F7SR3DN","schema_version":"1.0","canonical_sha256":"f97f28ec6d557317d7dcc345942d77b19524e043a5950a4cf032144a813ae914","source":{"kind":"arxiv","id":"1111.3266","version":2},"attestation_state":"computed","paper":{"title":"Bounds of a number of leaves of spanning trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anton Bankevich, Dmitri Karpov","submitted_at":"2011-11-14T16:19:56Z","abstract_excerpt":"We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\\over 4}(s-2)+2$ leaves.\n  Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\\ge 1$. We prove that $G$ has a spanning tree with at least $\\alpha_{g,k}(v(G)-k-2)+2$ leaves, where $\\alpha_{g,k}= {[{g+1\\over2}]\\over [{g+1\\over2}](k+3)+1}$ for $k<g-2$; $\\alpha_{g,k}= {g-2\\over (g-1)(k+2)}$ for $k\\ge g-2$.\n  We present infinite series of examples showing that all these bounds ar"},"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":"1111.3266","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-14T16:19:56Z","cross_cats_sorted":[],"title_canon_sha256":"8ba2e2e4d4fb8e03eeb072866a47eec628a57fc782f4d9a3d7806642f33bb6d1","abstract_canon_sha256":"5a5c8ccf0f95d4024ec47220c8cb450c06c63f04c7ff5ee62f8dadb46a5a7f6f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:59.783626Z","signature_b64":"5vnOcvMzbQ68IV872r3+jEgJgE3GxfT2+mKTPAZfgLk87A6gIdVPXmRzdCWQT37tgLHh9QwDtFyLD2mTGRWlDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f97f28ec6d557317d7dcc345942d77b19524e043a5950a4cf032144a813ae914","last_reissued_at":"2026-05-18T02:50:59.783158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:59.783158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds of a number of leaves of spanning trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anton Bankevich, Dmitri Karpov","submitted_at":"2011-11-14T16:19:56Z","abstract_excerpt":"We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\\over 4}(s-2)+2$ leaves.\n  Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\\ge 1$. We prove that $G$ has a spanning tree with at least $\\alpha_{g,k}(v(G)-k-2)+2$ leaves, where $\\alpha_{g,k}= {[{g+1\\over2}]\\over [{g+1\\over2}](k+3)+1}$ for $k<g-2$; $\\alpha_{g,k}= {g-2\\over (g-1)(k+2)}$ for $k\\ge g-2$.\n  We present infinite series of examples showing that all these bounds ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3266","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1111.3266","created_at":"2026-05-18T02:50:59.783249+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.3266v2","created_at":"2026-05-18T02:50:59.783249+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3266","created_at":"2026-05-18T02:50:59.783249+00:00"},{"alias_kind":"pith_short_12","alias_value":"7F7SR3DNKVZR","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"7F7SR3DNKVZRPV64","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"7F7SR3DN","created_at":"2026-05-18T12:26:22.705136+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/7F7SR3DNKVZRPV64YNCZILLXWG","json":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG.json","graph_json":"https://pith.science/api/pith-number/7F7SR3DNKVZRPV64YNCZILLXWG/graph.json","events_json":"https://pith.science/api/pith-number/7F7SR3DNKVZRPV64YNCZILLXWG/events.json","paper":"https://pith.science/paper/7F7SR3DN"},"agent_actions":{"view_html":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG","download_json":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG.json","view_paper":"https://pith.science/paper/7F7SR3DN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.3266&json=true","fetch_graph":"https://pith.science/api/pith-number/7F7SR3DNKVZRPV64YNCZILLXWG/graph.json","fetch_events":"https://pith.science/api/pith-number/7F7SR3DNKVZRPV64YNCZILLXWG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG/action/storage_attestation","attest_author":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG/action/author_attestation","sign_citation":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG/action/citation_signature","submit_replication":"https://pith.science/pith/7F7SR3DNKVZRPV64YNCZILLXWG/action/replication_record"}},"created_at":"2026-05-18T02:50:59.783249+00:00","updated_at":"2026-05-18T02:50:59.783249+00:00"}