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Frozen (Δ + 1)-colourings of bounded degree graphs

Published online by Cambridge University Press:  19 October 2020

Marthe Bonamy
Affiliation:
CNRS, LaBRI, Université de Bordeaux, France
Nicolas Bousquet
Affiliation:
LIRIS, CNRS, Université Claude Bernard Lyon 1, Lyon, France
Guillem Perarnau
Affiliation:
Departament de Matemàtiques (MAT), Universitat Politècnica de Catalunya (UPC), Barcelona, Spain
Corresponding
E-mail address:

Abstract

Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory 83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices.

In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n 2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

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Paper
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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