{"paper":{"title":"Nowhere-zero $3$-flow and $\\mathbb{Z}_3$-connectedness in Graphs with Four Edge-disjoint Spanning Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong-Jian Lai, Jiaao Li, Miaomiao Han","submitted_at":"2016-10-14T18:56:27Z","abstract_excerpt":"Given a zero-sum function $\\beta : V(G) \\rightarrow \\mathbb{Z}_3$ with $\\sum_{v\\in V(G)}\\beta(v)=0$, an orientation $D$ of $G$ with $d^+_D(v)-d^-_D(v)= \\beta(v)$ in $\\mathbb{Z}_3$ for every vertex $v\\in V(G)$ is called a $\\beta$-orientation. A graph $G$ is $\\mathbb{Z}_3$-connected if $G$ admits a $\\beta$- orientation for every zero-sum function $\\beta$. Jaeger et al. conjectured that every $5$-edge-connected graph is $\\mathbb{Z}_3$-connected. A graph is $\\langle\\mathbb{Z}_3\\rangle$-extendable at vertex $v$ if any pre-orientation at $v$ can be extended to a $\\beta$-orientation of $G$ for any ze"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04581","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"}