In the ABCD model, the interaction between a flavor field \(\Phi\) and the color fields \(A\), \(B\), \(C\), and \(D\) is of the form

This interaction gives rise to a sextic vertex with two flavor lines and four color lines.

It appears that the simplest process involving four-body forces requires six vertices (6-to-6, or 12-point). The six vertices can be arranged to form an octahedron. Since the octahedron has 8 faces, this is an 8-loop 6-body interaction.

It also appears that you cannot form simple generalizations of the octahedron with an odd number of bodies. But you can generalize the six-body octahedron to a \(2n\)-body \(n\)-antiprism. The \(n\)-antiprism has \(2n\) vertices, and \(2n+2\) faces, so this diagrams would correspond to \((2n+2)\)-loop \(2n\)-body interaction.

However, there appear to be special cases where you can also have vertex graphs that do not correspond to an antiprism. For example:

• 12-body interaction from a cuboctahedron graph
• 12-body interaction from a Chvátal graph
• 16-body interaction from a tesseract graph
• 16-body interaction from a Hoffman graph
• 20-body interaction from a Folkman graph
• 24-body interaction from a rhombicuboctahedron graph
• 30-body interaction from a icosidodecahedron graph
• 60-body interaction from a rhombicosidodecahedron graph

It is interesting that some of these graphs can also be realized as polyhedra. In principle, each of these graphs will produce a different many-body potential. For example, the cuboctahedron graph does not have hexagonal loop, like the 6-antiprism.

At this point it is not clear to me if there are more special cases. What I am looking for is a quartic graph with chromatic index equal to 4. That is, the graph only has quartic vertices (the external flavor lines are implicit) and each vertex takes four different kinds of edges.