Switching is a local transformation that when applied to a combinatorial object gives another object with the same parameters. It is here shown that the cycle switching graph of the 11 084 874 829 isomorphism classes of Steiner triple systems of order 19 as well as the cycle switching graph of the 1 348 410 350 618 155 344 199 680 000 labeled such designs are connected. In addition to giving an understanding of the multitude of Steiner triple systems—at least for order 19 but perhaps also generally—this work also presents an algorithm for testing connectedness of large implicit graphs and brings forward a benchmark instance for such algorithms.
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