Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S\_n$ and generating set the set of minimal transposition generators of the group $S\_n$, that is the set $S=\{(12),(13),\ldots, (1n)\}$. Then, will the graph be always planar?


The graph is easily seen to be bipartite with one part the set of odd permutations and the other part the set of even permutations. Upto the group $S\_3$ it is trivial to see why $G$ is planar. The first nontrivial part is when the group is $S\_4$. The degree is $3$ in this case, but I don't see the complete regular bipartite graph $K\_{3,3}$ explicitly. But can $G$ have a $K\_{3,3}$ minor? I don't think so. Also, what if I replace the set $S$ with the set of all transpositions? Any hints? thanks beforehand.



Transposition Cayley graphs are planar
You already have an answer regarding the first part of your question, but this uses the fact that with your given generating set, the Cayley graph is $(n-1)$-regular. What if we pick the generating set $\{ (12), (12\cdots n)\}$? 


So: there is a full characterisation, due to Maschke (1896), of which finite groups admit planar Cayley graphs with respect to some generating set. Such groups are called *planar*.



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> The finite group $A$ is planar if and only if $A = B\_1 \times B\_2$ with $B\_1 = 1$ or $\mathbb{Z}\_2$, and $B\_2 \in \{ \mathbb{Z}\_n, D\_n, S\_4, A\_4, A\_5 \mid n \in \mathbb{N} \}$. 
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A proof can be found e.g. [here](https://www.degruyter.com/view/title/121384?language=en), and the original article is the (aptly) named [Maschke, H.
*The Representation of Finite Groups, Especially of the Rotation Groups of the Regular Bodies of Three-and Four-Dimensional Space, by Cayley's Color Diagrams.*
Amer. J. Math. 18 (1896), no. 2, 156–194.]. If you'd like to read it, a stable link is [here](https://www.jstor.org/stable/2369680?seq=1#metadata_info_tab_contents). 


In particular, $S\_n$ for $n \geq 5$ admits *no* generating set such that its Cayley graph is planar, answering the second part of your question. 


