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

Abstract Given a finite family of graphs, we say that a graph is “‐free” if does not contain any graph in as a subgraph. We abbreviate ‐free to just “‐free” when . A vertex‐colored graph is called “rainbow” if no two vertices of have the same color. Given an integer and a finite family of graphs , let denote the smallest integer such that any properly vertex‐colored ‐free graph having contains an induced rainbow path on vertices. Scott and Seymour showed that exists for every complete graph . A conjecture of N. R. Aravind states that . The upper bound on that can be obtained using the methods of Scott and Seymour setting are, however, super‐exponential. Gyárfás and Sárközy showed that . For , we show that and therefore, . 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 , where . Moreover, in each case, our results imply the existence of at least distinct induced rainbow paths on vertices. Along the way, we obtain some new results on an oriented variant of the Gyárfás–Sumner conjecture. For , let denote the orientations of in which one vertex has out‐degree or in‐degree . We show that every ‐free oriented graph having a chromatic number at least and every bikernel‐perfect oriented graph with girth having a chromatic number at least contains every oriented tree on at most vertices as an induced subgraph.