Consider an undirected network $G$ with $n!$ vertices in it$,$ in which each

vertex corresponds to a permutation of $\{1,$dots,$n\}.$ We connect two vertices

with an undirected edge $(u,v)$ if the permutation associated with $u$ can

be transformed into the permutation associated with $v$ if we exchangeShow that an undirected network $G=(V,E)$ is bipartite if and only if it

does not contain any odd cycles.

a pair of adjacent elements; for example, the permutation $(14253)$ can

be transformed into $(12453)$ by exchanging the $2$ and the $4.$ Note that

we would not make an edge between $($for example$)(14253)$ and $(15243),$

because we would have to exchange the $4$ and the $5,$ which are not adjacent

to each other.

Show that $G$ is connected. If you like, you are welcome to assume that the

first entry and the last entry are also adjacent $($although $G$ is connected

either way$).$