vertices starting in this vertex. To see this, revisit the proof and let $P$ be the longest rainbow path starting in $v$, and observe that the argument made for Proposition \[prop:maximality\] still applies.

Rotations
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The paths in $K_n$ admit many symmetries; in fact, every permutation of the vertices of a path results in another path. We consider a special kind of permutation, which we call rotation here. Rotations were already used by Pósa in [@Posa1976]. For every $i\in[k]$, there is a rotation $\rho_i$ which acts on the path $P = (p_1,\dotsc,p_k)$ to produce the path $$\rho_i \cdot P = (p_i,p_{i-1},\dotsc,p_1,p_{i+1},p_{i+2},\dotsc,p_k)\text,$$ as shown in figure \[fig:rotation\].

(-5.2,-2)(5.2,2) (-5,0)[.08]{}[p1]{} (0,0)[.08]{}[pi]{} (1,0)[.08]{}[psi]{} (5,0)[.08]{}[pt]{}

(-2.75,0)[a]{} (-2.25,0)[b]{}

(2.75,0)[a1]{} (3.25,0)[b1]{}

The point is that if $P = (p_1,\dotsc,p_k)$ is a rainbow path and $p_i\in A(P)$, then $\rho_i\cdot P$ is a rainbow path that does not use the color $c(p_i,p_{i+1})$, but that is still very similar to $P$. In particular, $\rho_i\cdot P$ ends in the same vertex as $P$.

\[prop:maximality-rot\] If $P = (p_1,\dotsc,p_k)$ is a maximum rainbow path with respect to some coloring $c$, then for every $p_i\in A(P)$ we have $c(p_i,p_{i+1}){\not\in}c(p_k,V(P)^c)$. Similarly, for every $p_i\in B(P)$ we have $c(p_i,p_{i-1}) {\not\in}c(p_1,V(P)^c)$.

Suppose that $p_i\in A(P)$ 