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Title: Improper colourings of graphs
Author: Kang, Ross J.
ISNI:       0000 0001 3594 5686
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2008
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We consider a generalisation of proper vertex colouring of graphs, referred to as improper colouring, in which each vertex can only be adjacent to a bounded number t of vertices with the same colour, and we study this type of graph colouring problem in several different settings. The thesis is divided into six chapters. In Chapter 1, we outline previous work in the area of improper colouring. In Chapters 2 and 3, we consider improper colouring of unit disk graphs -- a topic motivated by applications in telecommunications -- and take two approaches, first an algorithmic one and then an average-case analysis. In Chapter 4, we study the asymptotic behaviour of the improper chromatic number for the classical Erdos-Renyi model of random graphs. In Chapter 5, we discuss acyclic improper colourings, a specialisation of improper colouring, for graphs of bounded maximum degree. Finally, in Chapter 6, we consider another type of colouring, frugal colouring, in which no colour appears more than a bounded number of times in any neighbourhood. Throughout the thesis, we will observe a gradient of behaviours: for random unit disk graphs and "large" unit disk graphs, we can greatly reduce the required number of colours relative to proper colouring; in Erdos-Renyi random graphs, we do gain some improvement but only when t is relatively large; for acyclic improper chromatic numbers of bounded degree graphs, we discern an asymptotic difference in only a very narrow range of choices for t.
Supervisor: McDiarmid, Colin J. H. Sponsor: NSERC (Canada) ; Commonwealth Scholarship Commission ; ORSAS
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Combinatorics ; Computer science (mathematics) ; Discrete mathematics (statistics) ; graph colouring ; probabilistic combinatorics ; random graphs ; unit disk graphs ; telecommunications ; improper colouring ; acyclic colouring ; frugal colouring