Negative correlation of edges in the uniform random forest of complete bipartite and complete multipartite graphs

A conjecture of Kahn and of Winkler, open since 2000, asserts that any two distinct edges of a finite graph are negatively correlated under the uniform measure on its forests. We prove two cases. First, for every complete bipartite graph K_{a,b} with a, b ≥ 2, every pair of distinct edges is strictly negatively correlated; for min(a, b) = 1 every pair gives equality. Two-variable Lagrange inversion expresses the forest counts through rescaled Laguerre polynomials, and each type of edge pair reduces to the sign of a concave quadratic in one ratio. This ratio is confined to an interval on which each quadratic is positive, with no asymptotic regime and no exceptional cases. Second, for every complete multipartite graph, with any number of parts and any part sizes, two edges that share a vertex are negatively correlated. The proof introduces the vertex polynomial Σ_S F(G/S) t^S, where F(G/S) counts the forests of G with the vertex set S merged into one vertex. For complete multipartite graphs this polynomial, after a fixed shift, is the image of the vertex-weighted matching polynomial under a linear operator whose symbol has no zeros on the product of upper half-planes. It is therefore real stable by the Heilmann–Lieb theorem and the Borcea–Brändén characterisation, and Brändén’s Rayleigh criterion gives the inequality. We conjecture that the vertex polynomial is real stable for every graph, which would give negative correlation of adjacent edges in all graphs, and report checks on all 1,244 graphs with 3 to 7 vertices and at least one edge. Every algebraic step not resting on a cited theorem is verified in Lean 4 with Mathlib.