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

Date and Time: Thursday, November 28, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Pascal Su

Chromatic Number of the Random Block Graph

The Erdős-Rényi random graph model was first introduced by Erdős and Rényi in the late 1950's and is still of great interest in the field of combinatorics. In particular we are interested in the chromatic number of a random graph. Bollobás proved the chromatic number is asymptotically \chi(G_n,1/2) ~ n/( 2 log₂ n ), for general constants p in (0,1) we now know \chi(G_n,p) ~ (log₂ (1/(1-p)) n )/(2 log₂ n) and there also exists work on lower order terms. The G_n,p model has many useful properties, but in some instances it might be too homogeneous and not flexible enough. We want to look at the random block graph model which can be viewed as a finer partition of the G_n,p random graph model. A quick description is partition the vertex set into a constant amount of sets, all of linear size, and place a random graph on every set and random bipartite graphs between every pair of sets and each of the probabilities p can vary. The chromatic number of this graph can be bounded, with a coupling, from above and below with \chi(G_{n, p_{min}}) and \chi(G_{n, p_{max}}) respectively. We prove the leading term for the chromatic number of the random block graph. Joint work with Anders Martinsson, Konstantinos Panagiotou and Miloš Trujić.

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