Register allocation and instruction scheduling are antagonistic optimizations: Whichever is applied rst, it will impede the other. To solve this problem, we propose dependence-conscious colouring, a register allocation method that takes the dependence graph used by the instruction schedu-ler into consideration. Dependence-conscious colouring consists of two parts: First, the interference graph ...

The Minimum Sum Colouring Problem is an NP-hard problem derived from the well-known graph colouring problem. It consists in finding a proper colouring which minimizes the sum of the assigned colours rather than the number of those colours. This problem often arises in scheduling and resource allocation. Mainly incomplete approaches were proposed, but Integer Linear Programming (ILP) and Constra...

A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a g...

A colouring of a graph G is called distinguishing if its stabiliser in AutG is trivial. It has been conjectured that, if every automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We study properties of random 2-colourings of locally finite graphs and show that the stabiliser of such a colouring is almost surely nowhere dense in AutG...

Vertex Descent is a local search algorithm which forms the basis of a wide spectrum of tabu search, simulated annealing and hybrid evolutionary algorithms for graph colouring. These algorithms are usually treated as experimental and provide strong results on established benchmarks. As a step towards studying these heuristics analytically, an analysis of the behaviour of Vertex Descent is provid...

We introduce a novel representation for the graph colouring problem, called the Integer Merge Model, which aims to reduce the time complexity of graph colouring algorithms. Moreover, this model provides useful information to aid in the creation of heuristics that can make the colouring process even faster. It also serves as a compact definition for the description of graph colouring algorithms....

In this paper, we consider a new edge colouring problem motivated by wireless mesh networks optimization: the proportional edge colouring problem. Given a graph G with positive weights associated to its edges, we want to find a proper edge colouring which assigns to each edge at least a proportion (given by its weight) of all the colours. If such colouring exists, we want to find one using the ...

For a positive integer k and graph G = (V,E), a k-colouring of G is a mapping c : V → {1, 2, . . . , k} such that c(u) 6= c(v) whenever uv ∈ E. The k-Colouring problem is to decide, for a given G, whether a k-colouring of G exists. The k-Precolouring Extension problem is to decide, for a given G = (V,E), whether a colouring of a subset of V can be extended to a k-colouring of G. A k-list assign...

A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree∆ has a strong edge-colouring with at most 4∆ + 4 colours. We show that 3∆ + 1 colours suffice if the graph has girth 6, and 4∆ colours suffice if ∆ ≥ 7 or the girth is at least 5. In the last part of the paper, we raise some questions ...