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...

Given graphs F and G and a nonnegative integer k, a function n : V(F) ~ {1 . . . . . k} is a G k-colouring of F if no induced copy of G is monochromatic; F is G k-chromatic if F has a G k-colouring but no G (k 1)-colouring. Further, we say F is uniquely G k-colourable if F is G k-chromatic and, up to a permutation of colours, it has only one G k-colouring. Such notions are extensions of the wel...

In this paper two best known at the moment algorithms for finding the maximum clique is compared. A key technique that both algorithms rely on is using a heuristic vertex colouring. The only difference between those algorithms is a way of using vertex-colouring and this difference in the very important. The algorithms’ approaches are divided in this paper into two classes: reapplying vertex col...

In this paper we propose a self-organizing method for the graph colouring problem. The proposed self-organizing method for the graph colouring extends the chemical casting model and makes it possible to find the number of colours needed to colour the graph. The self-organizing approach is needed to solve scheduling tasks in the applications of ad hoc networks. Nodes in ad hoc networks are usual...

The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs P1, . . . ,Pk (usually describing some common graph properties), is to decide, for a given graph G, whether the vertex set of G can be partitioned into sets V1, . . . , Vk such that, for each i, the induced subgraph of G on Vi belongs to Pi. It can be seen that GCOL generalizes many natural colouri...

The paper is focused on an implementation of a predictive controller with a colouring filter C in a disturbance model. Both single-input-single-output (SISO) and multi-input-multi-output (MIMO) cases were considered and analysed. The filter is often essential for practical applications of predictive control based on input-output models. It is commonly considered as a design parameter because it...

This paper considers the problem of online graph colouring. An algorithm for vertex-colouring graphs is said to be online if each vertex is irrevocably assigned a colour before later vertices are considered. We first introduce the problem formally and define a perfomance metric to evaluate the success of an online colouring algorithm. Then, several upper and lower bounds are presented on the pe...