Variance of random greedy independent sets in triangle-free graphs

This preprint studies the size of the independent set produced by processing the vertices of a fixed finite graph in uniformly random order, accepting each vertex when none of its neighbors has previously been accepted. For every finite simple triangle-free graph G with n ≥ 2 vertices and e edges, we prove the sharp variance bound Var(X_G) ≤ e((n − 2)/n)², with equality precisely for edgeless graphs and connected stars. Consequently, among trees of a given order n, the star uniquely maximizes the variance, attaining (n − 1)(n − 2)²/n². The proof combines expected vertex-deletion stability, a refined centered first-choice estimate, and a triangle-free edge-count identity through the law of total variance. The contribution is the sharp finite-order refinement of the elementary bound Var(X_G) ≤ e, together with the equality classification. Complete proofs are included, with no maximum-degree restriction. The corresponding variance inequality for arbitrary graphs remains unresolved in this work.