Abstract
Let \(\mathcal G\) be an infinite family of connected graphs and let k be a positive integer. We say that k is 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 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 collapses for \(\mathcal G\). 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.
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Caro, Y., Yuster, R. The Effect of Local Majority on Global Majorityin Connected Graphs. Graphs and Combinatorics 34, 1469–1487 (2018). https://doi.org/10.1007/s00373-018-1938-0
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DOI: https://doi.org/10.1007/s00373-018-1938-0