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Algebraic methods in Graph Theory; the linear algebra method, graph homomorphisms, group theoretic methods (for example Cayley graphs), and graph invariants. For graph eigenvalue problems use the spectral-graph-theory tag. For strongly regular graphs use the strongly-regular-graph tag. For Kneser graphs use the kneser-graph tag.
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A question about (unicity of certain cycles in a Cayley graph of a) symmetric group
The smallest $n$ for which there exist sequences as asked for is $n = 7$:
$(1,2,3,4,5,6,7) \cdot (1,2) \cdot (1,7,6,5,4,3,2) \cdot (1,2)
\cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot$
$(1,7,6,5,4,3, …