Lemma: For any two isomorphic graphs G and G$,$ if G has a circuit of length k then G$$ also has a circuit of length k$.$ More specifically, if

p: V$($G$)->$ V$($G$)$

and

q: E$($G$)->$ E$($G$)$

are the one$-$to$-$one correspondences that preserve the edge$-$endpoint functions of G and G$$ and if

C $=$ v$0$e$0$v$1$e$1$ ek $1$vk

is a circuit of length k in G$,$ then

C$=$ p$($v$0)$q$($e$0)$p$($v$1)$q$($e$1)$ q$($ek $1)$p$($vk$)$

is a circuit of length k in G$.$