Abstract
The conjecture was made by Kahn that a spanning forest F chosen uniformly at random from all forests of any finite graph G has the edge-negative association property. If true, the conjecture would mean that given any two edges ε1 and ε2 in G, the inequality \({{\mathbb{P}(\varepsilon_{1} \in \mathbf{F}, \varepsilon_{2} \in \mathbf{F}) \leq \mathbb{P}(\varepsilon_{1} \in \mathbf{F})\mathbb{P}(\varepsilon_{2} \in \mathbf{F})}}\) would hold. We use enumerative methods to show that this conjecture is true for n large enough when G is a complete graph on n vertices. We derive explicit related results for random trees.
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Stark, D. The Edge Correlation of Random Forests. Ann. Comb. 15, 529–539 (2011). https://doi.org/10.1007/s00026-011-0104-7
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DOI: https://doi.org/10.1007/s00026-011-0104-7