1 research outputs found
Uniformly Random Colourings of Sparse Graphs
We analyse uniformly random proper -colourings of sparse graphs with
maximum degree in the regime . This regime
corresponds to the lower side of the shattering threshold for random graph
colouring, a paradigmatic example of the shattering threshold for random
Constraint Satisfaction Problems. We prove a variety of results about the
solution space geometry of colourings of fixed graphs, generalising work of
Achlioptas, Coja-Oghlan, and Molloy on random graphs, and justifying the
performance of stochastic local search algorithms in this regime. Our central
proof relies only on elementary techniques, namely the first-moment method and
a quantitative induction, yet it strengthens list-colouring results due to Vu,
and more recently Davies, Kang, P., and Sereni, and generalises
state-of-the-art bounds from Ramsey theory in the context of sparse graphs. It
further yields an approximately tight lower bound on the number of colourings,
also known as the partition function of the Potts model, with implications for
efficient approximate counting