Chromatic number is not tournament-local

Scott and Seymour conjectured the existence of a function f: N -> Nsuch that, for every graph Gand tournament T on the same vertex set, chi(G) >= f(k) implies that chi(G[N-T(+)(v)]) > k for some vertex v. In this note we disprove this conjecture even if vis replaced by a vertex set of size O(log vertical bar V(G)vertical bar). As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomasse, and Wu concerning the corresponding statement where the graph Gis 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. (c) 2024 The Authors. Published by Elsevier Inc.