{"paper":{"title":"The Unit Acquisition Number of a Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anna Raleigh, Douglas B. West, Frederick Johnson, Paul S. Wenger","submitted_at":"2017-11-07T19:25:51Z","abstract_excerpt":"Let $G$ be a graph with nonnegative integer weights. A {\\it unit acquisition move} transfers one unit of weight from a vertex to a neighbor that has at least as much weight. The {\\it unit acquisition number} of a graph $G$, denoted $a_u(G)$, is the minimum size that the set of vertices with positive weight can be reduced to via successive unit acquisition moves when starting from the configuration in which every vertex has weight $1$.\n  For a graph $G$ with $n$ vertices and minimum degree $k$, we prove $a_u(G)\\le (n-1)/k$, with equality for complete graphs and $C_5$. Also $a_u(G)$ is at most t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02696","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"}