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Title: Connectivity properties of some transformation graphs
Author: Trakultraipruk, Somkiat
ISNI:       0000 0004 2738 7794
Awarding Body: London School of Economics and Political Science (University of London)
Current Institution: London School of Economics and Political Science (University of London)
Date of Award: 2013
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Many combinatorial problems can be formulated as "can we transform configuration 1 into configuration 2 if certain transformations are allowed?" In order to study such questions, we introduce a so-called transformation graph. This graph has the set of all possible configurations as its vertex set, and there is an edge between two configurations if one configuration can be obtained from the other by one of the allowed transformations. Then a question like "can we go from one configuration to another one" becomes a question about connectivity properties of transformation graphs. In this thesis, we study the following types of transformation graphs in particular: Labelled Token Graphs: Here configurations are arrangements of labelled tokens on a given graph, and we can go from one arrangement to another one by moving one token at a time along an edge of the given graph. We classify all cases when labelled token graphs are connected, and classify all pairs of arrangements that are in the same component. We also look at the problem how hard it is to determine the length of the shortest path between two arrangements. Strong k-Colour Graphs: For this transformation graph, the configurations are the proper vertex-colourings of a given graph with k colours, in which all k colours are actually used. We call such a colouring a strong k-colouring. We study the problem when we can transform any strong k-colouring into any other one by recolouring one vertex at a time, always maintaining a strong k-colouring. For certain classes of graphs, we can completely determine when the transformation graph of strong k-colourings is connected.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics