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Negative Correlation in Graphs and Matroids

The following two conjectures arose in the work of Grimmett and Winkler, and Pemantle: the uniformly random forest F and the uniformly random connected subgraph C of a finite graph G have the edge-negative association property. In other words, for all distinct edges e and f of G, the probability that F (respectively, C) contains e conditioned on containing f is less than or equal to the probability that F (respectively, C) contains e. Grimmett and Winkler showed that the first conjecture is true for all simple graphs on 8 vertices and all graphs on 9 vertices with at most 18 edges. In this paper, we describe an infinite, nontrivial class of graphs and matroids for which a generalized version of both conjectures holds.