Cycle Switching in Steiner Triple Systems of Order 19

Abstract

Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order $v$ (an $\text{STS}\left(v\right)$), yielding another $\text{STS}\left(v\right)$. This relationship may be represented by an undirected graph. An $\text{STS}\left(v\right)$ admits cycles of lengths $4,6,\dots ,v-7$ and $v-3$. In the particular case of $v=19$, it is known that the full switching graph, allowing the switching of cycles of any length, is connected. We show that if we restrict switching to only one of the possible cycle lengths, in all cases, the switching graph is disconnected (even if we ignore those $\text{STS}\left(19\right)$s, which have no cycle of the given length). Moreover, in a number of cases we find intriguing connected components in the switching graphs, which exhibit unexpected symmetries. Our method utilizes an algorithm for determining connected components in a very large implicitly defined graph which is more efficient than previous approaches, avoiding the necessity of computing canonical labelings for a large proportion of the systems.