An Inextensible Association Scheme Associated with a 4-regular Graph

Abstract.

 The scheme associated with a graph is an association scheme iff 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. If a presuperscheme associated with a graph is of height t, then the graph satisfies the (t+3)-vertex condition. On the other hand, the current paper provides an example of a regular and 3-regular graph, satisfying the 4-vertex condition, whose association scheme cannot be extended to a presuperscheme of height 1.