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 = { F }. 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 ℓ ( s , ℱ ) denote the smallest integer such that any properly vertex‐colored ℱ‐free graph G having χ ( G ) ≥ ℓ ( s , ℱ ) contains an induced rainbow path on s vertices. Scott and Seymour showed that ℓ ( s , K ) exists for every complete graph K. A conjecture of N. R. Aravind states that ℓ ( s , C 3 ) = s. The upper bound on ℓ ( s , C 3 ) that can be obtained using the methods of Scott and Seymour setting K = C 3 are, however, super‐exponential. Gyárfás and Sárközy showed that ℓ ( s , { C 3 , C 4 } ) = O ( ( 2 s ) 2 s ). For r ≥ 2, we show that ℓ ( s , K 2 , r ) ≤ ( r − 1 ) ( s − 1 ) ( s − 2 ) ∕ 2 + s and therefore, ℓ ( s , C 4 ) ≤ 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 ℓ ( s , { C 3 , C 4 , … , C g − 1 } ) ≤ s 1 + 4 g − 4, where g ≥ 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 ≥ 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 ( r − 1 ) ( s − 1 ) ( s − 2 ) + 2 s + 1 and every bikernel‐perfect oriented graph with girth g ≥ 5 having a chromatic number at least 2 s 1 + 4 g − 4 contains every oriented tree on at most s vertices as an induced subgraph.