<s>[INST] A question about (unicity of certain cycles in a Cayley graph of a) symmetric group Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-1}=(1,2,\ldots,n)\overbrace{(1,2)\cdots(1,2,\ldots,n)}^{2n-2}(1,2)=(1).$$ We want to know whether or not there exists another sequence of elements  $a_1,a_2,\ldots,a_{2n-4}\in S$ such that $$(12\ldots n)a_{2n-4}a_{2n-5}\cdots a_2a_1(12)=(1),$$ where  $a_{i+1}\neq a_i^{-1}$ for $i=0,1,2,\ldots,2n-4$ (putting $a_0=(12)$, $a_{2n-3}=(1,2,\ldots,n)$).
Equivalently, I want to ask if, in the cubic Cayley graph $Cay(S_n,S)$, there is a unique cycle of length $2(n-1)$ passing through $(1)$, $(1,2,\ldots,n)$ and $(1,2)$.
 [/INST] The smallest $n$ for which there exist sequences as asked for is $n = 7$:


*

*$(1,2,3,4,5,6,7) \cdot (1,2) \cdot (1,7,6,5,4,3,2) \cdot (1,2)
     \cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot$ 
$(1,7,6,5,4,3,2) \cdot (1,2) \cdot (1,2,3,4,5,6,7) \cdot (1,2)
     \cdot (1,7,6,5,4,3,2) \cdot (1,2) = ()$, and

*$(1,2,3,4,5,6,7) \cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot (1,7,6,5,4,3,2)
     \cdot (1,7,6,5,4,3,2) \cdot$
$(1,2) \cdot (1,2,3,4,5,6,7) \cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot
     (1,7,6,5,4,3,2) \cdot$
$(1,7,6,5,4,3,2) \cdot (1,2) = ()$.
For $n = 8$ there is no such sequence other than the trivial one
mentioned in the question, for $n = 9$ there is one, for $n = 10$ there
are $18$, for $n = 11$ there are $5$ and for $n = 12$ there are $104$ such
sequences. This has been found with the following GAP function:
SearchXueyiSequences := function ( n )

  local  sequences, search, S;

  search := function ( sequence, a )

    local  b;

    if Length(sequence) = 2*n-3 then
      if sequence[2*n-3] <> (1,2) and Product(sequence)*(1,2) = () then
        Add(sequences,Concatenation(sequence,[(1,2)]));
      fi;
      return;
    fi;
    for b in Difference(S,[a^-1]) do
      search(Concatenation(sequence,[a]),b);
    od;    
  end;

  S := Set(GeneratorsAndInverses(SymmetricGroup(n)));
  sequences := [];
  search([],S[2]);
  sequences := Set(sequences);
  sequences := sequences{[2..Length(sequences)]}; # exclude trivial sequence
  return sequences; 
end;

</s>