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

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

Clearly, any edge $\{p_k,r\}$ with $r\in V(P)^c$ cannot use the color of $\{p_{i+1},p_i\}$, as otherwise the path $$(p_i,\dotsc,p_1,p_{i+1},\dotsc,p_k,r)$$ would be a rainbow path on $\abs{V(P)}+1$ vertices, contradicting the choice of $P$. Viewed the other way around, we can say that a certain number of edges in $E(P)$ are not allowed to have colors in $c(p_k,V(P)^c)$. But all the edges in $E(p_k,V(P)^c)$ must be old, so their colors appear somewhere on the path. As Gyárfás and Mhalla observed, this conflict leads to the bound $k\geq (2n+1)/3$.

But what if we knew that starting in any vertex $r\in V(P)^c$, there is a rainbow path in $V(P)^c$ (of a certain minimum length $l$) that uses no colors of $c(P)$? Assume that this is the case, and, moreover, that given two arbitrary new colors, this path can be chosen in such a way that it does not use any one of them. Then instead of forbidding colors in $c(p_k,V(P)^c)$ to appear only on edges $\{p_i,p_{i+1}\}$ as above, we can also forbid them to appear on any edge $\{p_i,p_{i+1}\}$ such that $p_{i+1}$ has an $l$-successor $p_j$ with $c(p_1,p_j){\not\in}c(P)$. While this does not immediately lead to a good bound on the length 