A vertex colouring of a graph $G$ is "nonrepetitive" if contains no path for which the first half assigned same sequence colours as second half. Thue's famous theorem says that every nonrepetitively 3-colourable. This paper surveys results about nonrepetitive colourings graphs. The goal to give unified and comprehensive presentation major proof methods, well highlight numerous open problems.

We consider a vertex colouring of a connected plane graph G. A colour c is used k times by a face α of G if it appears k times along the facial walk of α. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such a colouring can be done using three c...

We prove several results about the complexity of the role colouring problem. A role colouring of a graph G is an assignment of colours to the vertices of G such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with 1 < k < n colours is NP-hard for planar graphs. We show that restricting the problem ...

Different voice qualities tend to vary in terms of their intrinsic loudness. Perceptual experiments have shown that voice quality variation can be strongly associated with the affective colouring of an utterance. The question addressed in this paper concerns the role that the intrinsic loudness variation might play in this voice quality-to-affect mapping. To test the hypothesis that the intrins...

A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For examp...

Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal colouring be reached from any given proper through a series of Kempe changes? We answer this question in affirmative for triangle-free graphs.

Abstract. Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximising the number of different colours available at the nodes and their neighbourhoods, given a predetermined number of colours. In the analytical framework of a tree approximation, carried out at both zero and finite temperatures, solutio...

We consider channel allocation to mitigate interference between wireless LANs. The channel allocation task is often formulated in the literature as finding a proper colouring of a single graph. We show that the interference between WLANs can be channel dependent in which case a different conflict graph is associated with each channel. Channel allocation then corresponds to a multi-graph colouri...