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

Date and Time: Tuesday, June 07, 2016, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Annika Heckel (University of Oxford)

The chromatic number of dense random graphs

The chromatic number of the random graph G(n,p) has been an intensely studied subject since at least the 1970s. A celebrated breakthrough by Bollobás in 1987 first established the asymptotic value of the chromatic number of G(n,1/2), and a considerable amount of effort has since been spent on refining Bollobás' approach, resulting in increasingly accurate bounds. Despite this, up until now there has been a gap of size O(1) in the denominator between the best known upper and lower bounds for the chromatic number of dense random graphs G(n,p) where p is constant.

In this talk, I will present new upper and lower bounds which match each other up to a term of size o(1) in the denominator. Somewhat surprisingly, the behaviour of the chromatic number changes around p=1-1/e^2, with a different limiting effect being dominant below and above this value. In contrast to earlier results, the upper bound was obtained through the second moment method, and I will give details on some aspects of the proof.

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