Semi-symmetric graph

A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two vertex sets of the bipartition (in fact, regularity is not required for this property to hold).

For instance, in the diagram of the Folkman graph shown here, green vertices can not be mapped to red ones by any automorphism, but every two vertices of the same color are symmetric with each other.

Semi-symmetric graphs were first studied E. Dauber, a student of F. Harary, in a paper, no longer available, titled "On line- but not point-symmetric graphs".

[1] The term "semi-symmetric" was first used by Klin et al. in a paper they published in 1978.

It was proven to be the smallest cubic semi-symmetric graph by Dragan Marušič and Aleksander Malnič.