{"paper":{"title":"A combinatorial property of flows on a cycle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhuo Diao","submitted_at":"2018-08-30T04:52:55Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10119","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"}