{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:RNW677EXZXBJMJ26V7X57VXS7L","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":"d57c018ab1fc383e7a32d146c1ecdea7d8cc63b2539a3fbc42b281ee642841b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-30T04:52:55Z","title_canon_sha256":"cfedb181d33e80a374eed336f7824762c8d3e48c294de5e5f955a02445db8efc"},"schema_version":"1.0","source":{"id":"1808.10119","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.10119","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"arxiv_version","alias_value":"1808.10119v1","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.10119","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"pith_short_12","alias_value":"RNW677EXZXBJ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RNW677EXZXBJMJ26","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RNW677EX","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:9aa73fc2f64c409c311ab0a0f2c2ad5126cda4fb0d9189eb2eafa1194bdbc48a","target":"graph","created_at":"2026-05-18T00:06:49Z","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":"In this paper, we prove a combinatorial property of flows on a cycle. $C(V,E)$ is an undirected cycle with two commodities: $\\{s_{1},t_{1}\\}, \\{s_{2},t_{2}\\}$;$r_1>0,r_2>0, \\mathbf r=(r_i)_{i=1,2}$ and $f,f'$ are both feasible flows for $(C,(s_i,t_i)_{i=1,2},\\mathbf r)$. Then $\\exists i\\in\\{1,2\\}, p\\in P_i, f(p)>0, \\forall e\\in p, f(e)\\geq f'(e)$ ; Here for each $i\\in\\{1,2\\}$, let $P_i$ be the set of $s_i$-$t_i$ paths in $C$ and $P=\\cup_{i=1,2}P_i$. This means given a two-commodity instance on a cycle, any two distinct network flow $f$ and $f'$, compared with $f'$, $f$ can't decrease every pat","authors_text":"Zhuo Diao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-30T04:52:55Z","title":"A combinatorial property of flows on a cycle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10119","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:b24e01bb36f5e98525d2989800947fda07aaa93f8af6dfce06d390eee4e76dc6","target":"record","created_at":"2026-05-18T00:06:49Z","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":"d57c018ab1fc383e7a32d146c1ecdea7d8cc63b2539a3fbc42b281ee642841b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-30T04:52:55Z","title_canon_sha256":"cfedb181d33e80a374eed336f7824762c8d3e48c294de5e5f955a02445db8efc"},"schema_version":"1.0","source":{"id":"1808.10119","kind":"arxiv","version":1}},"canonical_sha256":"8b6deffc97cdc296275eafefdfd6f2faccd2565338e5e469eb831ff9500b0835","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b6deffc97cdc296275eafefdfd6f2faccd2565338e5e469eb831ff9500b0835","first_computed_at":"2026-05-18T00:06:49.762598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:49.762598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IQG3dKYx3SNgjnQcRQA4axfBV5RRgQl4Oyc5CgG7envsLRvnoFPcnlXoShYMmFcZVHx0qUyhY6wGeIKYi2wFAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:49.763246Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.10119","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b24e01bb36f5e98525d2989800947fda07aaa93f8af6dfce06d390eee4e76dc6","sha256:9aa73fc2f64c409c311ab0a0f2c2ad5126cda4fb0d9189eb2eafa1194bdbc48a"],"state_sha256":"a7de72239bf72472da324be42d916d99d857b7023980e378015812dd155ddda9"}