Planting colourings silently
Victor Bapst, Amin Coja-Oghlan, Charilaos Efthymiou

TL;DR
This paper proves a strong concentration result for the number of proper k-colorings in random graphs, extending previous work and showing that the distribution is tightly concentrated around its expectation for a wide range of degrees.
Contribution
It establishes a significantly stronger concentration of the number of k-colorings in random graphs, valid up to the condensation phase transition, and demonstrates contiguity with the planted model.
Findings
Strong concentration of Z_k(G(n,m)) around its expectation.
Valid for a wide range of average degrees up to the phase transition.
Shows contiguity between uniform coloring and the planted model.
Abstract
Let be a fixed integer and let be the number of -colourings of the graph . For certain values of the average degree, the random variable is known to be concentrated in the sense that converges to in probability [Achlioptas and Coja-Oghlan: FOCS 2008]. In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, converges to in probability for any diverging function . For exceeding a certain constant this result covers all average degrees up to the so-called condensation phase transition, and this is best possible. As an application, we show that the experiment of choosing a -colouring of the random graph uniformly at random…
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