## Matchings, factors and cycles in graphs

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 (University of Nottingham only) (PhD) Woodall, Douglas R. Hamilton cycle, Ore-type condition, neighbourhood condition, neighborhood condition, Alon-Tarsi method, edge colouring, edge coloring Q Science > QA Mathematics > QA150 Algebra UK Campuses > Faculty of Science > School of Mathematical Sciences 10530 EP, Services 23 Jun 2008 20 Dec 2017 11:51 https://eprints.nottingham.ac.uk/id/eprint/10530

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