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Title: On edge colouring, fractionally colouring and partitioning graphs
Authors: Edwards, Katherine
Advisors: Seymour, Paul D
Contributors: Computer Science Department
Keywords: edge colouring
fractional colouring
graph colouring
graph theory
Subjects: Mathematics
Computer science
Issue Date: 2016
Publisher: Princeton, NJ : Princeton University
Abstract: We present an assortment of results in graph theory. First, Tutte conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. This generalizes the four-colour theorem. Robertson et al. had previously shown that to prove Tutte’s conjecture, it was enough to prove it for doublecross graphs. We provide a proof of the doublecross case. Seymour conjectured the following generalization of the four-colour theorem. Every d-regular planar graph can be d-edge-coloured, provided that for every odd-cardinality set X of vertices, there are at least d edges with exactly one end in X. Seymour’s conjecture was previously known to be true for values of d≤7. We provide a proof for the case d=8. We then discuss upper bounds for the fractional chromatic number of graphs not containing large cliques. It has been conjectured that each graph with maximum degree at most ∆ and no complete subgraph of size ∆ has fractional chromatic number bounded below ∆ by at least a constant f(∆). We provide the currently best known bounds for f(∆), for 4 ≤ ∆ ≤ 103. We also give a general upper bound for the fractional chromatic number in terms of the sizes of cliques and maximum degrees in local areas of a graph. Next, we give a result that says, roughly, that a graph with sufficiently large treewidth contains many disjoint subgraphs with ‘good’ linkedness properties. A similar result was a key tool in a recent proof of a polynomial bound in the excluded grid theorem. Our theorem is a quantitative improvement with a new proof. Finally, we discuss the p-centre problem, a central NP-hard problem in graph clustering. Here we are given a graph and an integer p, and need to identify a set of p vertices, called centres, so that the maximum distance from a vertex to its closest centre (the p-radius) is minimized. We give a quasilinear time approximation algorithm to solve p-centres when the hyperbolicity of the graph is fixed, with a small additive error on the p-radius.
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Computer Science

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