Philpotts, Adam Richard (2008) *Matchings, factors and cycles in graphs.* PhD thesis, University of Nottingham.

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## Abstract

A matching in a graph is a set of pairwise nonadjacent edges, a k-factor is a k-regular spanning subgraph, and a cycle is a closed path.

This thesis has two parts. In Part I (by far the larger part) we study sufficient conditions for structures involving matchings, factors and cycles. The three main types of conditions involve: the minimum degree; the degree sum of pairs of nonadjacent vertices (Ore-type conditions); and the neighbourhoods of independent sets of vertices. We show that most of our theorems are best possible by giving appropriate extremal graphs.

We study Ore-type conditions for a graph to have a Hamilton cycle or 2-factor containing a given matching or path-system, and for any matching and single vertex to be contained in a cycle. We give Ore-type and neighbourhood conditions for a matching L of l edges to be contained in a matching of k edges (l < k). We generalise two different aspects of this result: the l = 0 case with an Ore-type condition for a heavy matching in an edge-weighted graph; and the conditions for a perfect matching containing L with degree and neighbourhood conditions for a k-factor (k > 2) containing a given set of edges. We also establish neighbourhood conditions for the existence of a cycle of length at least k.

A list-edge-colouring of a graph is an assignment of a colour to each edge from its own list of colours. In Part II we study edge colourings of powers of cycles, and prove the List-Edge-Colouring Conjecture for squares of cycles of odd length.

Item Type: | Thesis (PhD) |
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Supervisors: | Woodall, Douglas R. |
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Uncontrolled Keywords: | Hamilton cycle, Ore-type condition, neighbourhood condition, neighborhood condition, Alon-Tarsi method, edge colouring, edge coloring |
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Faculties/Schools: | UK Campuses > Faculty of Science > School of Mathematical Sciences |
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ID Code: | 530 |
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Deposited By: | Adam Richard Philpotts |
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Deposited On: | 23 Jun 2008 |
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Last Modified: | 06 Feb 2009 14:43 |
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