We consider finite colourings of products $$X_1\times X_2\times \cdot \times X_n$$ infinite sets and determine what is the minimal number colours a subproduct $$Y_1\times Y_2\times Y_n$$ subsets could achieve.

Combinatorial optimization is a way of finding an optimum solution from a finite set of objects. For combinatorial optimization problems, the number of possible solutions grows exponentially with the number of objects. These problems belong to the class of NP hard problems for which probably efficient algorithm does not exist. Using the distributed approach with parallelization these problems c...

The application of the probabilistic method to graph colouring has been yielding interesting results for more than 40 years. Several probabilistic tools are presented in this survey, ranging from the basic to the more advanced. For each of them, an application to a graph colouring problem is presented in detail. In this way, not only is the general idea of the method exposed, but also are the c...

In 1993 Voloshin introduced the concept of mixed hypergraph. A mixed hypergraph is characterised by the fact that it possesses anti-edges as well as edges. In a colouring of a mixed hypergraph, every anti-edge has at least two vertices of the same colour. In this paper a new parameter is introduced: the monochromatic block number mb. It is the number of monochromatic blocks in a colouring of a ...

12 A vertex colouring of a 2-connected plane graph G is a strong parity 13 vertex colouring if for every face f and each colour c, the number of 14 vertices incident with f coloured by c is either zero or odd. 15 Czap et al. in [9] proved that every 2-connected plane graph has a 16 proper strong parity vertex colouring with at most 118 colours. 17 In this paper we improve this upper bound for s...

Let the edges of a graph G be coloured so that no colour is used more than k times. We refer to this as a k-bounded colouring. We say that a subset of the edges of G is multicoloured if each edge is of a different colour. We say that the colouring is H-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ARk = {G : every k-bounded colouring of G is H-good}...

A non-degenerate rectangular subdivision is a subdivision of a rectangle into a set of non-overlapping rectangles S, such that no four rectangles meet in a point. We consider a problem that Katz and colleagues call strong polychromatic four-colouring: Colouring the vertices of the subdivision with four colours, such that each rectangle of S has all colours among its four corners. By considering...

We introduce a notion of probability to the graph programming language GP2 which resolves nondeterministic choices of graph transformation rules and their matches. With our programming model Probabilistic GP2 (P-GP2), rule and match decisions are assigned uniform distributions over their domains. In this paper, we present an implementation of P-GP2 as an extension of an existing GP2 compiler. A...

A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χc(G) ≤ ∆ ∗+2 for any 3-connected plane graph G with maximum face degree ∆∗. It is known that the...

We describe a greedy vertex colouring method which can be used to colour optimally the edge set of certain chordal graphs. This new heuristic yields an exact edge-colouring algorithm for odd maximumdegree doubly chordal graphs. This method shows that any such graph G can be edge-coloured with maximum degree (G) colours, i.e., all these graphs are Class 1. In addition, this method gives a simple...