A proof of the Alon–Seymour–Thomas conjecture

This preprint presents a proof of the Alon–Seymour–Thomas conjecture on balanced separators in graphs excluding a complete graph as a minor. More precisely, it establishes that for every integer h ≥ 3, every n-vertex graph containing no K_h minor admits a vertex set S of size at most min{n, 21h√n} such that every connected component of G − S has at most 2n/3 vertices. The argument develops a connected hitting-set theorem using vertex expansion, finite optimal transport duality, and averaged transport estimates. This theorem is incorporated into the separator framework of Alon, Seymour, and Thomas, together with subsequent refinements. The manuscript was developed with AI assistance under the author's guidance, as disclosed in the paper.