Bounding the chromatic number of dense digraphs by arc neighborhoods
Felix Klingelhoefer, Alantha Newman
TL;DR
The paper tackles bounding the directed chromatic number $\vec{\chi}$ of dense digraphs by examining arc neighborhoods. It develops an arc-local-to-global methodology for tournaments via $t$-arc-boundedness, Jewels, and jewel-chains, then extends the approach to dense digraphs with bounded independence number using a recursive dense function $\mathrm{dense}(t,\alpha)$. A key outcome is that local bounds on arc neighborhoods imply global bounds on $\vec{\chi}$ for both tournaments and general dense digraphs, with a concrete base case $\mathrm{dense}(t,1)=f(t)$ where $f$ comes from the tournament analysis. As an application, the authors prove the equivalence of the El-Zahar–Erdős conjecture and a Nguyen–Scott–Seymour conjecture regarding high-chromatic structures, thereby linking graph and tournament colorings. The work integrates structural tools (dominant/absorbing sets, clusters, Jewels) with a careful arc-decomposition to bridge local neighborhood structure and global coloring in dense directed graphs.
Abstract
The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc $uv$ in a tournament $T$ is the set of vertices that form a directed triangle with arc $uv$. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erdős and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.