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Coloring Cubic Graphs by Point-Intransitive Steiner Triple Systems



An math formula-coloring of a cubic graph G is an edge coloring of G by points of a Steiner triple system math formula such that the colors of any three edges meeting at a vertex form a block of math formula. 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.

  1. We give a sufficient condition for a Steiner triple system math formula to be universal.
  2. 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.
  3. 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).