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 that f (k) ≤ (k − 1)^2 and conjectured f (k) = 2n − 2. In this paper, we give some sufficient conditions for an n-chromatic digraphs to contains some oriented tree. In particular, we show that every acyclic n-chromatic digraph contains every oriented tree of order n. We also show that f (k) ≤ k^2/2 − k/2 + 1. Finally, we consider the existence of antidirected trees in digraphs. We prove that every antidirected tree of order k is containedinevery(5k−9)-chromaticdigraph. Weconjecturethatif|E(D)|>(k−2)|V(D)|,thenthedigraph D contains every antidirected tree of order k. This generalizes Burr's conjecture for antidirected trees and the celebrated Erdo ̋s-So ́s Conjecture. We give some evidences for our conjecture to be true.

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