Oriented trees in digraphs

Description

Let $f(k)$ be the smallest integer such that every $f(k)$-chromatic digraph contains every oriented tree of order $k$. Burr proved $f(k)\leq (k-1)^2$ in general, and he conjectured $f(k)=2k-2$. Burr also proved that every $(8k-7)$-chromatic digraph contains every antidirected tree. We improve both of Burr's bounds. We show that $f(k)\leq k^2/2-k/2+1$ and that every antidirected tree of order $k$ is contained in every $(5k-9)$-chromatic digraph. We make a conjecture that explains why antidirected trees are easier to handle. It states that if $|E(D)| > (k-2) |V(D)|$, then the digraph $D$ contains every antidirected tree of order $k$. This is a common strengthening of both Burr's conjecture for antidirected trees and the celebrated Erd\H{o}s-Sós Conjecture. The analogue of our conjecture for general trees is false, no matter what function $f(k)$ is used in place of $k-2$. We prove our conjecture for antidirected trees of diameter 3 and present some other evidence for it. Along the way, we show that every acyclic $k$-chromatic digraph contains every oriented tree of order $k$ and suggest a number of approaches for making further progress on Burr's conjecture.

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