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

TL;DR

The paper advances Burr's conjecture by establishing a subquadratic universal bound for oriented trees in digraphs with large chromatic number. It introduces gluing lemmas for leaves and directed/oriented paths, forming a win-win induction that either attaches many leaves or decomposes into few directed paths to grow the target tree while controlling the bound. The main results include a subquadratic universal bound of $8 \sqrt{\frac{2}{15}} k\sqrt{k} + \frac{11}{3}k + \sqrt{\frac{5}{6}} \sqrt{k} + 1$ for general oriented trees, an improved arborescence bound of $\sqrt{\frac{4}{3}} k \sqrt{k} + k/2$, and a linear bound $(b-1)(k-3)+3$ for $b$-block paths ($b\ge 2$). These methods provide a framework that combines structural decompositions with constructive path-gluing to push towards tighter universality results and illuminate directions for further improvements.

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 $b\ge 2$.

Figures (2)

• Figure 1: The partition of $V(D)$ into sets $X',Y',Z'$. The next step of the induction yields partition $X,Y,Z$, shown in grey, green, and red respectively. DAGs $D[Y']$ and $D[Y]$ are layered such that all arcs go from left to right. The sinks $S'$ of $D[Y']$ begin a directed path of length $\ell$ continuing in $Z'$, in dash-dotted. Then, the sinks $S$ of $D[Y]$ begin a directed path of length $\ell+1$ continuing in $Z$.
• Figure 2: The partition of $V(D)$ into sets $X',Y',Z'$. The next step of the induction yields partition $X,Y,Z$, shown in grey, green, and red respectively. Vertices of $Y'$ begin a copy of $Q'$, in dash-dotted. Then, vertices of $Y$ begin a copy of $Q$, either in $K'$ (see $y$), or by appending an arc to some $Q'$ (see $x'$).

Theorems & Definitions (16)

• Conjecture 1: Burr burrConjecture
• Theorem 2: Addario-Berry et al. addario-berryOrientedTreesDigraphs2013
• Theorem 2
• Theorem 3: Addario-Berry et al. addario-berryPathsTwoBlocks2007
• Theorem 3
• Lemma 4
• Lemma 5: Addario-Berry et al. addario-berryOrientedTreesDigraphs2013
• Corollary 6
• Lemma 7
• Lemma 8