Oriented trees in digraphs

Abstract

International audienceLet $f(k)$ 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

Subdivisions of oriented cycles in digraphs with large chromatic number

“…This was slightly improved by Addario‐Berry et al. who proved the following. Theorem Every ‐chromatic oriented graph contains every oriented tree of order k .…”

Isometric copies of directed trees in orientations of graphs

“…In Burr considered the numbers and proved that . This upper bound was improved to the upper bound in . According to (still unproved) Conjecture of Burr , the equality holds for all .…”

χ‐bounded families of oriented graphs

“…This was slightly improved by Addario‐Berry et al. by replacing by . The right bound is conjectured to be .…”