We consider a simple randomised algorithm that seeks a weak 2colouring of a hypergraph H; that is, it tries to 2-colour the points of H so that no edge is monochromatic. If H has a particular well-behaved form of such a colouring, then the method is successful within expected number of iterations O(n3), when H has n points. In particular, when applied to a graph G with n nodes and chromatic num...

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theo...

The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no induced subgraph isomorphic to $H_1$ or $H_2$. A connected graph $H_1$ is almost classified if Colouring on $(H_1,H_2)$-free graphs is known to be polynomial-time ...

Approximate random k-colouring of a graph G = (V,E) is a very well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution, i.e. the uniform distribution over all the k-colourings of G. Here, we deal with the problem when the underlying graph is an instance of Erdős-Rényi random graph G(n, p), wh...

A colouring of a graph G = (V,E) is a mapping c : V → {1, 2, . . .} such that c(u) 6= c(v) if uv ∈ E; if |c(V )| ≤ k then c is a k-colouring. The Colouring problem is that of testing whether a given graph has a k-colouring for some given integer k. If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The complexity of Colouring for H-free g...

We consider n-dimensional Euclidean lattice networks with nearest neighbour coupling architecture. The associated lattice dynamical systems are infinite systems of ordinary differential equations, the cells, indexed by the points in the lattice. A pattern of synchrony is a finite-dimensional flow-invariant subspace for all lattice dynamical systems with the given network architecture. These sub...

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 ...

For many problems in Scheduling and Timetabling the choice of a mathematical programming formulation is determined by the formulation of the graph colouring component. This paper briefly surveys seven known integer programming formulations of vertex colouring and introduces a new formulation using “supernodes”. In the definition of George and McIntyre [SIAM J. Numer. Anal. 15 (1978), no. 1, 90–...