Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, October 12, 2023, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Raphael Steiner

Chromatic number is not tournament-local

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[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 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. Based on joint work with Girao, Illingworth, Hendrey, Lehner, Michel and Savery.

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