Chromatic number is not tournament-local

Abstract

Scott and Seymour conjectured the existence of a function $f:N\to N$ such that, for every graph G and tournament T on the same vertex set, $\chi \left(G\right)⩾f\left(k\right)$ implies that $\chi \left(G\right[N_T^{+}\left(v\right)\left]\right)⩾k$ for some vertex v. In this note we disprove this conjecture even if v is replaced by a vertex set of size $O(\log |V\left(G\right)\left|\right)$. As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomassé, and Wu concerning the corresponding statement where the graph G is replaced by another tournament, and disprove a related conjecture of Nguyen, Scott, and Seymour. We also show that the setting where chromatic number is replaced by degeneracy exhibits a quite different behaviour.