I need to prove that permutation graph is a perfect graph. I've shown that for any $i<j$, $\{i,j\}$ connected iff $f(i)>f(j)$. In addition, I know that the complement of a permutation graph is a permutation graph. Then, it's enough to prove it has no odd cycles $c(n)$ for $n>1$. I tried to prove by contradiction. Suppose $x_1,x_2,\ldots,x_{2n+1}$ are the vertices of the cycle. Now, it will be nice to assume that $x_1<x_2<x_3<...<x_{2n+1}$ as then we're done, but it's wrong, because we can have the permutation: $f:=(3,4,1,2)$ and we get $c(4)$ and if we assumed $x_1<x_2<x_3<x_4$ we got a contradiction.
Thank you in advance!