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

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

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Jozef Skokan (LSE)

Graphs with large minimum degree and large girth are three-colourable

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 {C₃, C₅} 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 {C₃, C₅}-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 {C₃, C₅, ... , C_2k-1} is 1/(2k-1). However, their proof does not imply a good bound on the chromatic number of {C₃, C₅, ... , 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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