Given a finite family of $\cal{F}$ graphs, $G$ subgraphs $\cal{ F}$. A vertex color graph \(H\) is called a “rainbow” if no two vertices in $H$ have the same color. Given an integer $s$ and a finite family of graphs $\cal{F}$, $\ell(s,\cal{F})$ is the appropriate vertex color $\cal{F $\chi(G The }$-free graph $G$ with )\geq \ell(s,\cal{F})$ contains the rainbow paths induced to the $s$ vertices. Scott and Seymour showed that for every complete graph $K$ there exists $\ell(s,K)$. N.~R.~Aravind’s conjecture states that $\ell(s,C_3)=s$. However, the upper bound on $\ell(s,C_3)$ that can be obtained using Scott and Seymour’s method of setting $K=C_3$ is hyperexponential. Gy\’arf\’as and S\’ark\”ozy are $\ell(s,\{C_3,C_4\})=\cal{O}\big((2s)^{2s}\big) for $.$r\geq 2$, showed $\ell(s,K_{2,r})\leq (r-1)(s-1)s/2+s$ , thus $\ell (s,C_4)\leq\frac{s^2+s}{2}$. This greatly improves the bounds of Gy\’arf\’as and S\’ark\”ozy , and larger classes of graphs are also covered. Adapt the proof to achieve stronger upper bounds for higher Garth graphs: $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq s^ {1+ \frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case our result is at least $s!/2$ on the $s$ vertex. , implying that there are distinct induced rainbow pathways for In the process, we obtain some new results on directed variants of the Gy\’arf\’as-Sumner conjecture. For $r\geq 2$, let $\cal{B}_r$ represent the direction of $K_{2,r}$ and one vertex has exterior degree $r$. All directed $\cal{B}_r$-free graphs have chromaticity of at least $(r-1)(s-1)(s-2)+2s+1$, and all bikernel fully directed graphs An enclosure $g\geq 5$ with at least $2s^{1+\frac{4}{g-4}}$ chromaticity contains all directed trees with at most $s$ vertices as induced subgraphs. increase.
