{"paper":{"title":"Min-Max Theorems for Packing and Covering Odd $(u,v)$-trails","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Chaitanya Swamy, Sharat Ibrahimpur","submitted_at":"2017-08-23T00:18:44Z","abstract_excerpt":"We investigate the problem of packing and covering odd $(u,v)$-trails in a graph. A $(u,v)$-trail is a $(u,v)$-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Let $\\nu(u,v)$ denote the maximum number of edge-disjoint odd $(u,v)$-trails, and $\\tau(u,v)$ denote the minimum size of an edge-set that intersects every odd $(u,v)$-trail.\n  We prove that $\\tau(u,v)\\leq 2\\nu(u,v)+1$. Our result is tight---there are examples showing that $\\tau(u,v)=2\\nu(u,v)+1$---and substantially improves upon the bound of $8$ obtaine"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.06857","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"}