Variants of the Gyárfás–Sumner conjecture: Oriented trees and rainbow paths

Given a finite family $ℱ$ of graphs, we say that a graph $G$ is “$ℱ$-free” if $G$ does not contain any graph in $ℱ$ as a subgraph. We abbreviate $ℱ$-free to just “$F$-free” when $ℱ=\left\{F\right\}$. 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 $ℱ$, let $\ell \left(s,ℱ\right)$ denote the smallest integer such that any properly vertex-colored $ℱ$-free graph $G$ having $\chi \left(G\right)\ge \ell \left(s,ℱ\right)$ contains an induced rainbow path on $s$ vertices. Scott and Seymour showed that $\ell \left(s,K\right)$ exists for every complete graph $K$. A conjecture of N. R. Aravind states that $\ell \left(s,C_3\right)=s$. The upper bound on $\ell \left(s,C_3\right)$ that can be obtained using the methods of Scott and Seymour setting $K=C_3$ are, however, super-exponential. Gyárfás and Sárközy showed that $\ell \left(s,\left\{C_3,C_4\right\}\right)=O\left({\left(2s\right)}^{2s}\right)$. For $r\ge 2$, we show that $\ell \left(s,K_{2,r}\right)\le \left(r-1\right)\left(s-1\right)\left(s-2\right)∕2+s$ and therefore, $\ell (s,C_4)\le \frac{s^2-s+2}{2}$. This significantly improves Gyárfás and Sárközy'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 $\ell \left(s,\left\{C_3,C_4,\dots ,C_{g-1}\right\}\right)\le s^{1+\frac{4}{g-4}}$, where $g\ge 5$. 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 new results on an oriented variant of the Gyárfás–Sumner conjecture. For $r\ge 2$, let ${ℬ}_r$ denote the orientations of $K_{2,r}$ in which one vertex has out-degree or in-degree $r$. We show that every ${ℬ}_r$-free oriented graph having a chromatic number at least $\left(r-1\right)\left(s-1\right)\left(s-2\right)+2s+1$ and every bikernel-perfect oriented graph with girth $g\ge 5$ having a chromatic number at least $2s^{1+\frac{4}{g-4}}$ contains every oriented tree on at most $s$ vertices as an induced subgraph.