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arXiv · 2609.18175

A near-linear upper bound for Burr's conjecture

Liangdong Fan·Junying Lu·Yaojun Chen
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Abstract

Let $f(k)$ denote the smallest integer such that every oriented graph $D$ with chromatic number at least $f(k)$ contains every oriented tree on $k$ vertices. Burr (1980) showed that $f(k)\le (k-1)^2$ and conjectured that $f(k)=2k-2$. Bessy, Gonçalves and Reinald (2025) proved that $f(k)=O(k^{3/2})$. In this paper, by using an absorbing set method, we show that $f(k)\le \lfloor 31\log (k!)\rfloor=O(k\log k)$.

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Liangdong Fan, Junying Lu, Yaojun Chen. 2026-09-16. A near-linear upper bound for Burr's conjecture. https://arxiv.org/abs/2609.18175

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