Oriented trees in digraphs

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)≤(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....

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Bibliographic Details
Published in:Discrete mathematics 2013-04, Vol.313 (8), p.967-974
Main Authors: Addario-Berry, Louigi, Havet, Frédéric, Linhares Sales, Cláudia, Reed, Bruce, Thomassé, Stéphan
Format: Article
Language:English
Subjects:
Analogue
Burrs
Chromatic number
Computer Science
Discrete Mathematics
Graph theory
Integers
Mathematical analysis
Oriented trees
Strengthening
Trees
Unavoidable digraph
Universal digraph
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Summary: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)≤(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)≤k2/2−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 Erdős-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.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2013.01.011