# Oriented trees in digraphs

2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : 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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Journal articles

Cited literature [25 references]

https://hal.inria.fr/hal-00821609
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Submitted on : Sunday, October 23, 2016 - 4:07:49 PM
Last modification on : Monday, February 11, 2019 - 4:40:02 PM

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### Citation

Louigi Addario-Berry, Frédéric Havet, Claudia Linhares Sales, Bruce Reed, Stéphan Thomassé. Oriented trees in digraphs. Discrete Mathematics, Elsevier, 2013, 313 (8), pp.967-974. ⟨http://www.sciencedirect.com/science/article/pii/S0012365X13000289⟩. ⟨10.1016/j.disc.2013.01.011⟩. ⟨hal-00821609⟩

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