{"paper":{"title":"Berge's Conjecture and Aharoni-Hartman-Hoffman's Conjecture for locally in-semicomplete digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C\\^andida Nunes da Silva, Carla Negri Lintzmayer, Maycon Sambinelli, Orlando Lee","submitted_at":"2017-08-22T16:07:52Z","abstract_excerpt":"Let $k$ be a positive integer and let $D$ be a digraph. A path partition $\\sP$ of $D$ is a set of vertex-disjoint paths which covers $V(D)$. Its $k$-norm is defined as $\\sum_{P \\in \\sP} \\Min{|V(P)|, k}$. A path partition is $k$-optimal if its $k$-norm is minimum among all path partitions of $D$. A partial $k$-coloring is a collection of $k$ disjoint stable sets. A partial $k$-coloring $\\sC$ is orthogonal to a path partition $\\sP$ if each path $P \\in \\sP$ meets $\\min\\{|P|,k\\}$ distinct sets of $\\sC$. Berge (1982) conjectured that every $k$-optimal path partition of $D$ has a partial $k$-colorin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.06691","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"}