An Adinkra is an N-regular, edge-N-colored graph satisfying the following square completion condition:
- For every vertex v and distinct edge colors a, b, we have abv = bav \neq v. In other words, the vertices v, av, bv can be completed into a square with sides of alternating colors a and b.
In addition, Adinkras come equipped with the following additional information:
- A bipartition of the vertices of the graph, labeled white and black, such that the two vertices adjacent to any edge are of opposite types. Equivalently, there is cubical cohomology cochain b \in C^0(A,\mathbb{Z}_2) with (db)(e) \equiv 1 for all edges e.
- An assignment of signs \pm 1 to the edges, with negative edges drawn with dashed lines, such that the product of the signs of the edges around any square is -1. Equivalently, there is a cubical cohomology cochain s \in C^1(A,\mathbb{Z}_2) with (ds)(f) \equiv 1 for all square faces f. Alternatively, one could replace edge signs with edge orientations as follows: a sign of +1 corresponds to an edge directed from white to black vertices, and a sign of -1 corresponds to an edge directed from black to white vertices. Such a structure is called a Kasteleyn orientation, which at least for surface graphs corresponds to a discrete spin structure.
- An assignment of an integral height to the vertices, such that the two vertices adjacent to any edge have heights differing by 1. Equivalently, there is a cubical cohomology cochain h \in C^0(A,\mathbb{Z}) with (dh)(e) = \pm 1 for all edges e. Furthermore, this integral cochain reduces to the bipartition cochain b modulo 2.