Cycle switching in Steiner triple systems of order 19

TL;DR

This paper investigates the structure of cycle switching graphs in Steiner triple systems of order 19, revealing that restricting cycle lengths disconnects the graph and uncovering surprising symmetries in its components.

Contribution

It demonstrates that limiting cycle lengths in switching operations disconnects the graph and introduces an efficient algorithm for analyzing large, implicitly defined graphs.

Findings

Switching graphs are disconnected when restricted to single cycle lengths.

Unexpected symmetries are observed in certain connected components.

An efficient algorithm improves analysis of large implicit graphs.

Abstract

Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order $v$ (an $STS(v)$), yielding another $STS(v)$. This relationship may be represented by an undirected graph. An $STS(v)$ admits cycles of lengths $4,6,\ldots,v-7$ and $v-3$. In the particular case of $v=19$, it is known that the full switching graph, allowing 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 $STS(19)$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 utilises 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 labellings for a large proportion of the systems.