Presuperschemes and Colored Directed Graphs

The scheme associated with a graph is an association scheme if and only if the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme in the case of a colored, directed graph. The obstacles can be expressed in terms of \(t\)-vertex conditions. If a graph does not satisfy the \(t\)-vertex condition, a prescheme associated with it cannot be erected beyond the \((t-3)\)rd-level.