Year: 2025
Type: article
Abstract: ABSTRACT Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order (an ), yielding another . This relationship may be represented by an undirected graph. An admits cycles of lengths and . In the particular case of , 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 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.
Source: Journal of Combinatorial Designs
DOI: 10.1002/jcd.21975
Authors: Grahame Erskine, Terry S. Griggs
Institution: The Open University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4408307774