Approximate random k-colouring of a graph G is a 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 in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdős-Rényi random graph G(n, d/n), where d is a sufficiently large constant. We propose a n...

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...

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...

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 ...

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...

Tomato is an important crop and hence there is a great interest in understanding the genetic basis of its flowering. Several genes have been identified by mutations and we constructed a set of novel double mutants to understand how these genes interact to shape the inflorescence. It was previously suggested that the branching of the tomato inflorescence depends on the gradual transition from in...

We present the first linear time algorithm for d-list colouring of a graph—i.e. a proper colouring of each vertex v by colours coming from lists L(v) of sizes at least deg(v). Previously, procedures with such complexity were only known for ∆-list colouring, where for each vertex v one has |L(v)| ≥ ∆, the maximum of the vertex degrees. An implementation of the procedure is available.

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

This paper describes a branch-and-cut procedure for an extension of the bounded colouring problem, generally known as curriculum-based university course timetabling. In particular, we focus on Udine Course Timetabling [di Gaspero and Schaerf, J. Math. Model. Algorithms 5:1], which has been used in Track 3 of the 2007 International Timetabling Competition. First, we present an alternative intege...