Edge-colorings of cubic graphs with elements of point-transitive Steiner triple systems

A cubic graph G is S-edge-colorable for a Steiner triple system S if its edges can be colored with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. We show that a cubic graph is S-edge-colorable for every non-trivial affine Steiner triple system S unless it contains a well-defined obstacle called a bipartite end. In addition, we show that all cubic graphs are S-edge-colorable for every non-projective non-affine point-transitive Steiner triple system S.