Chromatic number is not tournament-local

Abstract

Scott and Seymour conjectured the existence of a function $f:\mathbb{N}\to \mathbb{N}$ such that, for every graph $G$ and tournament $T$ on the same vertex set, $\ddot{I}‡(G)⩾f(k)$ implies that $\ddot{I}‡(G[N_T^{+}(v)])⩾k$ for some vertex $v$. In this note we disprove this conjecture even if $v$ is replaced by a vertex set of size $\mathcal{O}(\log |V(G)|)$. 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.

7 pages; funding information added