The Minimal Non-Koszul A(Gamma)

The algebras $A(\mathrm{\Gamma })$, where $\mathrm{\Gamma }$ is a directed layered graph, were first constructed by I. Gelfand, S. Serconek, V. Retakh and R. Wilson. These algebras are generalizations of the algebras $Q_n$, which are related to factorizations of non-commutative polynomials. It was conjectured that these algebras were Koszul. In 2008, T.Cassidy and B.Shelton found a counterexample to this claim, a non-Koszul $A(\mathrm{\Gamma })$ corresponding to a graph $\mathrm{\Gamma }$ with 18 edges and 11 vertices. We produce an example of a directed layered graph $\mathrm{\Gamma }$ with 13 edges and 9 vertices which produces a non-Koszul $A(\mathrm{\Gamma })$. We also show this is the minimal example with this property.