Unimodular polytopes and a new Heller-type bound on totally unimodular matrices via Seymour's decomposition theorem

We prove a sharp upper bound on the number of columns of a totally unimodular matrix with column sums $1$ which improves upon Heller's bound by a factor of $2$. The proof uses Seymour's decomposition theorem. Unimodular polytopes are lattice polytopes such that the vertices of every full-dimensional subsimplex form an affine lattice basis. This is a subclass of 0/1-polytopes and contains for instance edge polytopes of bipartite graphs. Our main result implies a sharp upper bound on the number of vertices of unimodular polytopes.