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 Theory83 (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(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

中文翻译：

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有界度图的冻结（Δ + 1）着色

让G成为一个图表n顶点和最大度数Δ，并让ķ是一个整数。这G的k-重新着色图是顶点为的图ķ- 着色G还有两个ķ-如果它们在一个顶点上完全不同，则它们是相邻的。众所周知，ķ-重新着色图连接用于$k\geq \Delta+2$. 费加利、约翰逊和保卢斯马（J. 图论83(2016) 340-358) 表明 (Δ + 1)-重新着色图由大小至少为 2 的唯一连通分量和（可能许多）孤立顶点组成。在本文中，我们研究孤立顶点的比例 (也叫冷冻（Δ + 1）重新着色图中的着色）。我们的第一个贡献是表明，如果G是连通的，冷冻色素的比例G在指数范围内更小n超过着色的总数。这促使使用 Glauber 动力学来近似图的 (Δ + 1) 着色的数量。与推测的混合时间相反○(n日志n） 为了$k\geq \Delta+2$颜色，我们证明了（Δ + 1）颜色的芒硝动力学的混合时间仅限于非冷冻颜色可以是Ω（n2）。最后，我们证明了一些关于大周长和冻结着色的图存在的结果，并研究了随机规则图中的冻结着色。