Rees-algebra formulation of the Caccetta-Häggkvist conjecture

Let $D$ be a digraph on $n$ vertices, let $G(D)$ be its associated bipartite graph, and let $M_D$ be the perfect matching associated with $D$. If $J$ is the defining ideal of the Rees algebra $R[I(G(D))t]$, Rees-algebra formulation of the Caccetta-Häggkvist conjecture. then, for some $q\leq\ell$, $J$ has a binomial generator of degree $q$ that is square-free and has relatively prime terms, one of which is a product of elements of $M_D$. The paper obtains this reformulation from the Jacobian-dual formulation using its theorem on the defining ideal of the Rees algebra; it is therefore equivalent to the original directed-cycle conjecture.