Chromatic number is not tournament-local
António Girão, Kevin Hendrey, Freddie Illingworth, Florian Lehner, Lukas Michel, Michael Savery, Raphael Steiner · Journal of Combinatorial Theory Series B · 2024
Scott and Seymour conjectured the existence of a function f:N→N such that, for every graph G and tournament T on the same vertex set, χ(G)⩾f(k) implies that χ(G[NT+(v)])⩾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(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.