The Effect of Local Majority on Global Majorityin Connected Graphs

Let G be an infinite family of connected graphs and let k be a positive integer. We say that k is forcing for G if for all G∈ 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 weakly forcing for G if any such weighing implies that the weight of G is bounded from below by a constant. Otherwise we say that kcollapses for G. We classify k for some of the most prominent classes of graphs, such as all connected graphs, all connected graphs with a given maximum degree and all connected graphs with a given average degree.