Frozen colourings of bounded degree graphs

Q2 Mathematics

Electronic Notes in Discrete Mathematics Pub Date : 2018-07-01 DOI:10.1016/j.endm.2018.06.029

Marthe Bonamy, Nicolas Bousquet, Guillem Perarnau

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引用次数: 3

Abstract

Let G be a graph with maximum degree Δ and k be an integer. The k-recolouring graph of G is the graph whose vertices are proper k-colourings of G and where two colourings are adjacent iff they differ on exactly one vertex. Feghali, Johnson and Paulusma showed that the ( $Δ + 1$ )-recolouring graph is composed by a unique connected component and (possibly many) isolated vertices, also known as frozen colourings of G.

Motivated by its applications to sampling, we study the proportion of frozen colourings of connected graphs. Our main result is that the probability a proper colouring is frozen is exponentially small on the order of the graph. The obtained bound is tight up to a logarithmic factor on Δ in the exponent. We briefly discuss the implications of our result on the study of the Glauber dynamics on ( $Δ + 1$ )-colourings. Additionally, we show that frozen colourings may exist even for graphs with arbitrary large girth. Finally, we show that typical Δ-regular graphs have no frozen colourings.

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有界度图的冻结着色

设G为最大次为Δ的图，k为整数。G的k重着色图是这样一个图，它的顶点是G的k适当着色，如果两个着色恰好在一个顶点上不同，那么它们是相邻的。Feghali, Johnson和Paulusma表明(Δ+1)-重着色图是由一个唯一的连接分量和(可能是许多)孤立的顶点组成的，也称为g的冻结着色。受其在采样中的应用的启发，我们研究了连接图的冻结着色比例。我们的主要结果是，一个适当的着色被冻结的概率在图的顺序上是指数小的。得到的边界紧绷到指数中Δ上的对数因子。我们简要地讨论了我们的结果对(Δ+1)-着色剂的Glauber动力学研究的意义。此外，我们还证明，即使对于任意大周长的图形，也可能存在冻结着色。最后，我们证明了典型的Δ-regular图没有固定的着色。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

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