  Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is
"$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a
subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of
$H$ have the same color. Given an integer $s$ and a finite family of graphs
$\mathcal{F}$, let $\ell(s,\mathcal{F})$ denote the smallest integer such that
any properly vertex-colored $\mathcal{F}$-free graph $G$ having
$\chi(G)\geq\ell(s,\mathcal{F})$ contains an induced rainbow path on $s$
vertices. Scott and Seymour showed that $\ell(s,K)$ exists for every complete
graph $K$. A conjecture of N. R. Aravind states that $\ell(s,C_3)=s$. The upper
bound on $\ell(s,C_3)$ that can be obtained using the methods of Scott and
Seymour setting $K=C_3$ are, however, super-exponential. Gy\'arf\'as and
S\'ark\"ozy showed that $\ell(s,\{C_3,C_4\})=\mathcal{O}\big((2s)^{2s}\big)$.
For $r\geq 2$, we show that $\ell(s,K_{2,r})\leq (r-1)(s-1)(s-2)/2+s$ and
therefore, $\ell(s,C_4)\leq\frac{s^2-s+2}{2}$. This significantly improves
Gy\'arf\'as and S\'ark\"ozy's bound and also covers a bigger class of graphs.
We adapt our proof to achieve much stronger upper bounds for graphs of higher
girth: we prove that $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq
s^{1+\frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case, our results
imply the existence of at least $s!/2$ distinct induced rainbow paths on $s$
vertices. Along the way, we obtain some results on related problems on oriented
graphs. For $r\geq 2$, let $\mathcal{B}_r$ denote the orientations of $K_{2,r}$
in which one vertex has out-degree or in-degree $r$. We show that every
$\mathcal{B}_r$-free oriented graph $G$ having $\chi(G)\geq
(r-1)(s-1)(s-2)+2s+1$ and every bikernel-perfect oriented graph $G$ with girth
$g\geq 5$ having $\chi(G)\geq 2s^{1+\frac{4}{g-4}}$ contains every $s$ vertex
oriented tree as an induced subgraph.
