Let γ : E(G) −→ N∗ be an edge colouring of a graph G and σγ : V (G) −→ N∗ the vertex colouring given by σγ(v) = ∑ e3v γ(e) for every v ∈ V (G). A neighbour-sumdistinguishing edge-colouring of G is an edge colouring γ such that for every edge uv in G, σγ(u) 6= σγ(v). The study of neighbour-sum-distinguishing edge-colouring of graphs was initiated by Karoński, Łuczak and Thomason [8]. They conjec...

We examine some flexible notions of constraint satisfaction, observing some relationships between model theoretic notions of universal Horn class membership and robust satisfiability. We show the NP-completeness of 2-robust monotone 1-in-3 3SAT in order to give very small examples of finite algebras with NP-hard variety membership problem. In particular we give a 3-element algebra with this pro...

We determine the limiting distribution of the logarithm of the number of satisfying assignments in the random k-uniform hypergraph 2-colouring problem in a certain density regime for all k > 3. As a direct consequence we obtain that in this regime the random colouring model is contiguous with respect to the planted model, a result that helps simplifying the transfer of statements between these ...

This paper describes an ant algorithm that colours graphs in an e cient way. First, we describe the graph colouring problem and the ant algorithm. We also present some results and propose a simple generalisation of the algorithm that might allow its application to other assignment problems. Finally, a short study of the algorithm operators explains its performance and shows that it may be seen ...

Many combinatorial problems can be formulated as “Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?”. An example of such a question is: given two k-colourings of a graph, can I transform the first k-colouring into the second one, by recolouring one vertex at a time, and always maintaining a proper k-colouring? Another example is: given two soluti...

Graph colouring and its generalizations are useful tools in modelling a wide variety of scheduling and assignment problems. In this paper we review several variants of graph colouring, such as precolouring extension, list colouring, multicolouring, minimum sum colouring, and discuss their applications in scheduling.

The graph colouring problem is a widely studied combinatorial optimisation problem with a highly symmetric solution space. Many heuristics and search algorithms are unattractive for general solution, i.e. they generate poor approximate solutions, because they were originally designed for specific problem instances. Population-based approaches like genetic algorithms do not provide a good altern...

For a positive integer k, a k-colouring of a graph G = (V,E) is a mapping c : V → {1, 2, . . . , k} such that c(u) 6= c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a k-colouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem k-COLOURING instead. We survey known results on the computational complexity of CO...

In graph theory, Graph Colouring Problem (GCP) is an assignment of colours to vertices of any given graph such that the colours on adjacent vertices are different. The GCP is known to be an optimization and NP-hard problem. Imperialist Competitive Algorithm (ICA) is a meta-heuristic optimization and stochastic search strategy which is inspired from socio-political phenomenon of imperialistic co...

Lovász invoked topological colouring bounds in proving Kneser’s Conjecture. Subsequently, numerous applications of topological techniques to graph colouring problems have arisen. However, even today, little is known about how to construct a graph with a particular topological colouring bound, or about the structure of such graphs. The aim of this thesis is to remedy this deficit. First, we will...