New Steiner 2-designs from old ones by paramodifications

Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals

Coloring cubic graphs by point-intransitive Steiner triple systems

An \cs-colouring of a cubic graph $G$ is an edge-colouring of $G$ by points of a Steiner triple system \cs such that the colours of any three edges meeting at a vertex form a block of \cs. A Steiner triple system which colours every simple cubic graph is said to be universal. It is known that every non-trivial point-transitive Steiner triple system that is neither projective nor affine is universal. In this paper we present the following results. (1) We give a sufficient condition for a Steiner triple system \cs 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 the last author in 2004