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arXiv · 2402.19351

Oriented trees in $O(k \sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture

Stéphane Bessy·Daniel Gonçalves·Amadeus Reinald
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Abstract

In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $\frac{k^2}{2} - \frac{k}{2} + 1$ by Addario-Berry et al. We give the first subquadratic bound for Burr's conjecture, by showing that every directed graph with chromatic number $8\sqrt{\frac{2}{15}} k \sqrt{k} + O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $\sqrt{\frac{4}{3}} k \sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $b\ge 2$.

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Stéphane Bessy, Daniel Gonçalves, Amadeus Reinald. 2024-02-29. Oriented trees in $O(k \sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture. https://arxiv.org/abs/2402.19351

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