Anagram-Free Colourings of Graphs

A sequenceSis calledanagram-freeif it contains no consecutive symbolsr1r2. . .rkrk+1. . .r2ksuch thatrk+1. . .r2kis a permutation of the blockr1r2. . .rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graphGis calledanagram-freeif the sequence of colours on any path inGis anagram-free. We call the minimal number of colours needed for such a colouring theanagram-chromaticnumber ofG.In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we essentially give each vertex a separate colour.

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Adjacency Labelling for Planar Graphs (and Beyond)

Journal of the ACM ◽

10.1145/3477542 ◽

2021 ◽

Vol 68 (6) ◽

pp. 1-33

Author(s):

Vida Dujmović ◽

Louis Esperet ◽

Cyril Gavoille ◽

Gwenaël Joret ◽

Piotr Micek ◽

...

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Order Term ◽

Optimal Result ◽

Induced Subgraph ◽

Bounded Degree ◽

Graph Classes ◽

Bounded Degree Graphs ◽

Minor Free Graphs ◽

Labelling Scheme

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n -vertex planar graph G is assigned a (1 + o(1)) log 2 n -bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G . This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n , there exists a graph U n with n 1+o(1) vertices such that every n -vertex planar graph is an induced subgraph of U n . These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k -planar graphs.

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

Combinatorics Probability Computing ◽

10.1017/s0963548320000139 ◽

2020 ◽

pp. 1-14

Author(s):

Marthe Bonamy ◽

Nicolas Bousquet ◽

Guillem Perarnau

Keyword(s):

Graph Theory ◽

Maximum Degree ◽

Mixing Time ◽

Glauber Dynamics ◽

Regular Graphs ◽

Connected Component ◽

Bounded Degree ◽

Large Girth ◽

Bounded Degree Graphs

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 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.

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