{"paper":{"title":"On the balanced decomposition number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Tadashi Sakuma","submitted_at":"2012-12-11T06:22:07Z","abstract_excerpt":"A {\\em balanced coloring} of a graph $G$ means a triple $\\{P_1,P_2,X\\}$ of mutually disjoint subsets of the vertex-set $V(G)$ such that $V(G)=P_1 \\uplus P_2 \\uplus X$ and $|P_1|=|P_2|$. A {\\em balanced decomposition} associated with the balanced coloring $V(G)=P_1 \\uplus P_2 \\uplus X$ of $G$ is defined as a partition of $V(G)=V_1 \\uplus \\cdots \\uplus V_r$ (for some $r$) such that, for every $i \\in \\{1,\\cdots,r\\}$, the subgraph $G[V_i]$ of $G$ is connected and $|V_i \\cap P_1| = |V_i \\cap P_2|$. Then the {\\em balanced decomposition number} of a graph $G$ is defined as the minimum integer $s$ suc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2308","kind":"arxiv","version":3},"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"}