{"paper":{"title":"The effect of local majority on global majority in connected graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Raphael Yuster, Yair Caro","submitted_at":"2017-11-26T17:04:09Z","abstract_excerpt":"Let ${\\mathcal G}$ be an infinite family of connected graphs and let $k$ be a positive integer. We say that $k$ is ${\\it forcing}$ for ${\\mathcal G}$ if for all $G \\in {\\mathcal G}$ but finitely many, the following holds. Any $\\{-1,1\\}$-weighing of the edges of $G$ for which all connected subgraphs on $k$ edges are positively weighted implies that $G$ is positively weighted. Otherwise, we say that it is ${\\it weakly~forcing}$ for ${\\mathcal G}$ if any such weighing implies that the weight of $G$ is bounded from below by a constant. Otherwise we say that $k$ ${\\it collapses}$ for ${\\mathcal G}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.09422","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"}