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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 "Proceedings of the 31st Workshop on GraphTheoretic Concepts in Computer Science (WG’05)", 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
Excluding Minors in Cubic Graphs
 Combin. Probab. Comput
, 1996
"... . Let P 10 ne be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P 10 ne. The decomposition is used to show that graphs in this class are 3edgecolourable. We also consider an application to a conjecture du ..."
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. Let P 10 ne be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P 10 ne. The decomposition is used to show that graphs in this class are 3edgecolourable. We also consider an application to a conjecture due to Grotzsch which states that a planar graph is 3edgecolourable if and only if it is fractionally 3edgecolourable. 1985 Mathematics Subject Classification: 05C50,05C75. Key Words and Phrases: Planar graph, Petersen Graph, Four Colour Problem, cubic, matching polyhedron, integer decomposition property, edgecolouring. 1 Background We consider loopless graphs G = (V; E) with node set V and edge set E. For an edge e 2 E, we denote by G=e the graph obtained by contracting the edge e, i.e., identifying its ends and deleting the resulting loop. A subgraph (respectively induced subgraph) of G is a graph obtained by deleting nodes or edges (respectively deleting nodes) of G. Informally, a...
Planarity Testing and Embedding
, 2004
"... Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 19 ..."
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Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 1930, an efficient solution to it was found only in the seventies of the last century. Planar graphs play an important role both in the graph theory and in the graph drawing areas. In fact, planar graphs have several interesting properties: for example they are sparse, fourcolorable, allow a number of operations to be performed efficiently, and their structure can be elegantly described by an SPQRtree (see Section 3.1.2). From the information visualization perspective, instead, as edge crossings turn out to be the main culprit for reducing readability, planar drawings of graphs are considered clear and comprehensible. As a matter of fact, the study of planarity has motivated much of the development of graph theory. In this chapter we review the number of alternative algorithms available in the literature for efficiently testing planarity and computing planar embeddings. Some of these algorithms
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 : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 ..."
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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 : : : : : : : : : : : : : : : : : : :
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