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...

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 ...

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...

The following EREW PRAM algorithms for edge-colouring a general graph are presented: 1. an algorithm that finds a ðDþ dÞ-edge-colouring, 1pdoD; in Oððlog d þ ðD=dÞÞ log nÞ time, using n þ m processors; 2. an algorithm that finds a D1þe-edge-colouring, 0oeo1; in OðlogD log nÞ time, using nD1þe processors. r 2003 Elsevier Inc. All rights reserved.

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 ...

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 generalised colouring numbers colr(G) and wcolr(G) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe e...

The oriented colouring problem is intuitive and related to undirected colouring, yet remains NP-hard even on digraph classes with bounded traditional directed width measures. Recently we have also proved that it remains NP-hard in otherwise severely restricted digraph classes. However, unlike most other problems on directed graphs, the oriented colouring problem is not directly transferable to ...

Variations and extensions of the basic vertex-colouring and edge-colouring models have been developed to deal with increasingly complex scheduling problems. We present and illustrate them in specific situations where additional requirements are imposed. We include list-colouring, mixed graph colouring, co-colouring, colouring with preferences, bandwidth colouring, and present applications of ed...