Abstract
Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at least one triangle. Triangle switches can be used to define Markov chains which generate graphs with a given degree sequence and with many more triangles (3-cycles) than is typical in a uniformly random graph with the same degrees. We show that the set of triangle switches connects the set of all d-regular graphs on n vertices, for all \(d \ge 3\). Hence, any Markov chain which assigns positive probability to all triangle switches is irreducible on these graphs. We also investigate this question for 2-regular graphs.
Dyer supported by EPSRC grant EP/S016562/1,“Sampling in hereditary classes”, Greenhill by Australian Research Council grant DP190100977.
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Cooper, C., Dyer, M., Greenhill, C. (2021). A Triangle Process on Regular Graphs. In: Flocchini, P., Moura, L. (eds) Combinatorial Algorithms. IWOCA 2021. Lecture Notes in Computer Science(), vol 12757. Springer, Cham. https://doi.org/10.1007/978-3-030-79987-8_22
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