Katherine Edwards will present her FPO "On edge colouring, fractionally colouring and partitioning graphs" on Tuesday, 5/31/2016 at 11am, 402 Computer Science Building. The members of her committee are Paul Seymour (adviser), Readers: Maria Chudnovsky (Math) and Chun-Hung Liu (Math); Nonreaders: Robert Tarjan and Bernard Chazelle. A copy of her thesis is available in Room 310. Everyone is invited to attend her talk. The abstract follow below. 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 double cross 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 su"ciently 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.