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On the Number of Solutions in Random Graph k-Colouring

Part of: Graph theory

Published online by Cambridge University Press:  21 May 2018

FELICIA RASSMANN
FELICIA RASSMANN*
Affiliation:
Institute of Mathematics, Goethe University, 10 Robert-Mayer-Straße, Frankfurt 60325, Germany (e-mail: rassmann@math.uni-frankfurt.de)
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Abstract

Let k ⩾ 3 be a fixed integer. We exactly determine the asymptotic distribution of ln Zk(G(n, m)), where Zk(G(n, m)) is the number of k-colourings of the random graph G(n, m). A crucial observation to this end is that the fluctuations in the number of colourings can be attributed to the fluctuations in the number of small cycles in G(n, m). Our result holds for a wide range of average degrees, and for k exceeding a certain constant k0 it covers all average degrees up to the so-called condensation phase transition.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 1 , January 2019 , pp. 130 - 158
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 278857–PTCC.

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