Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, December 06, 2022, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Jozef Skokan (LSE)

Let *F* be a fixed family of graphs. The homomorphism threshold of *F* is the infimum of those α for which there exists an *F*-free graph H=H(*F*, α), such that every *F*-free graph on *n* vertices of minimum degree *α n* is homomorphic to H.

Letzter and Snyder showed that the homomorphism threshold of the family {C_{3}, C_{5}} is 1/5. They also found explicit graphs H(*F*, α), for α> 1/5 + ε, which were in addition 3-colourable. Thus, their result also implies that {C_{3}, C_{5}}-free graphs of minimum degree at least (1/5 + ε)n are 3-colourable. For longer cycles, Ebsen and Schacht showed that the homomorphism threshold of {C_{3}, C_{5}, ... , C_{2k-1}} is 1/(2k-1). However, their proof does not imply a good bound on the chromatic number of {C_{3}, C_{5}, ... , C_{2k-1}}-free graphs of minimum degree n/(2k-1). Answering a question of Letzter and Snyder, we prove that such graphs are 3-colourable.

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