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Mutual Exclusion Scheduling
 Theoretical Computer Science
, 1996
"... Mutual exclusion scheduling is the problem of scheduling unittime tasks nonpreemptively on m processors subject to constraints represented by a graph G, such that tasks represented by adjacent vertices in G must run in disjoint time intervals. This problem arises in loadbalancing the parallel ..."
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Mutual exclusion scheduling is the problem of scheduling unittime tasks nonpreemptively on m processors subject to constraints represented by a graph G, such that tasks represented by adjacent vertices in G must run in disjoint time intervals. This problem arises in loadbalancing the parallel solution of partial differential equations by domain decomposition. Minimizing the completion time is NPhard even if either the number of processors or the completion time is fixed but greater than two. However, polynomial time is sufficient to produce optimal schedules for forests, and simple heuristics perform well on certain classes of graphs. For graphs derived from the twodimensional domain decomposition problem, heuristics yield solutions within 4c \Gamma 7 time units of optimal, where c is the maximal number of regions that touch each other at a single point in the domain decomposition; these solutions are within a constant factor of optimal. 1.
Channel assignment and improper choosability of graphs
 in &quot;Proceedings of the 31st Workshop on GraphTheoretic Concepts in Computer Science (WG’05)&quot;, Lecture Notes in Computer Science
, 2005
"... Abstract. We model a problem proposed by Alcatel, a satellite building company, using improper colourings of graphs. The relation between improper colourings and maximum average degree is underlined, which contributes to generalise and improve previous known results about improper colourings of plan ..."
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Abstract. We model a problem proposed by Alcatel, a satellite building company, using improper colourings of graphs. The relation between improper colourings and maximum average degree is underlined, which contributes to generalise and improve previous known results about improper colourings of planar graphs. 1
Graph Colouring via the Discharging Method
, 2003
"... In this thesis we study two colouring problems on planar graphs. The main technique we use is the Discharging Method, which was used to prove the Four Colour Theorem. The first ..."
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In this thesis we study two colouring problems on planar graphs. The main technique we use is the Discharging Method, which was used to prove the Four Colour Theorem. The first
MSM3P5: Graph theory Notes 1999
"... Contents 1 Introduction 5 1.1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.2 History of the fourcolour problem : : : : : : : : : : : : : : : : : : 6 2 Applications of Euler's formula 11 2.1 Euler's formula : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Contents 1 Introduction 5 1.1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.2 History of the fourcolour problem : : : : : : : : : : : : : : : : : : 6 2 Applications of Euler's formula 11 2.1 Euler's formula : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.2 Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3 Kempe's approach 19 3.1 The first `proof' of the fourcolour theorem : : : : : : : : : : : : : 19 3.2 The fivecolour theorem : : : : : : : : : : : : : : : : : : : : : : : 21 3.3 A reduction theorem : : : : : : : : : : : : : : : : : : : : : : : : : 22 4 Other approaches to the problem 25 4.1 Hamilton cycles : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4.2 Edgecolourings : : : : : : : : : : : : : : : : : : :