1.1 The Problem Let G = (V,E) be a simple undirected graph. A vertex colouring of G assigns colours to each vertex in such a way that neighbours have different colours. In this paper we discuss how efficient (time and bits) vertex colouring may be accomplished by exchange of bits between neighbouring vertices. The distributed complexity of vertex colouring is of fundamental interest for the stu...

Inflorescence brassicas, such as broccoli and cauliflower, rich in fiber vitamins, are widely consumed Brazil, present an average production of 23 t ha-1. Despite this demand, the quality seedlings both species is one main factors that limit production. Thus, objective study was to evaluate cauliflower seedling emergence different substrates, using irrigation method named Deep Film Technique (D...

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...

An acyclic edge colouring of a graph is a proper edge colouring having no 2-coloured cycle, that is, a colouring in which the union of any two colour classes forms a linear forest. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and is usually denoted by a′(G). Determining a ′(G) exactly is a very hard problem (both the...

The k-colouring reconfiguration problem asks whether, for a given graph G, two proper k-colourings α and β of G, and a positive integer `, there exists a sequence of at most ` proper k-colourings of G which starts with α and ends with β and where successive colourings in the sequence differ on exactly one vertex ofG. We give a complete picture of the parameterized complexity of the k-colouring ...

Ramsey theory is about results in the following style: no matter how “chaotic” the overall structure is, if we look at the right (usually small) piece of the structure, we will find a pattern. The most standard version of this type of result is the graph Ramsey theorem. Here, the structure we examine is a colouring of the edges of a graph (not necessarily a proper colouring), and the pattern we...

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 ...