An -coloring of a cubic graph G is an edge coloring of G by points of a Steiner triple system such that the colors of any three edges meeting at a vertex form a block of . A Steiner triple system that colors every simple cubic graph is said to be universal. It is known that every nontrivial point-transitive Steiner triple system that is neither projective nor affine is universal. In this article, we present the following results.
- We give a sufficient condition for a Steiner triple system to be universal.
- With the help of this condition we identify an infinite family of universal point-intransitive Steiner triple systems that contain no proper universal subsystem. Only one such system was previously known.
- We construct an infinite family of non-universal Steiner triple systems none of which is either projective or affine, disproving a conjecture made by Holroyd and Škoviera (J Combin Theory Ser B 91 (2004), 57–66).