Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.559858 
Title:  Cycles in edgecoloured graphs and subgraphs of random graphs  
Author:  White, M. D. 
ISNI:
0000 0004 2728 9174


Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  2011  
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Abstract:  
This thesis will study a variety of problems in graph theory. Initially, the focus will be on finding minimal degree conditions which guarantee the existence of various subgraphs. These subgraphs will all be formed of cycles, and this area of work will fall broadly into two main categories. First to be considered are cycles in edgecoloured graphs and, in particular, two questions of Li, Nikiforov and Schelp. It will be shown that a 2edgecoloured graph with minimal degree at least 3n/4 either is isomorphic to the complete 4partite graph with classes of order n/4, or contains monochromatic cycles of all lengths between 4 and n/2 (rounded up). This answers a conjecture of Li, Nikiforov and Schelp. Attention will then turn to the length of the longest monochromatic cycle in a 2edgecoloured graph with minimal degree at least cn. In particular, a lower bound for this quantity will be proved which is asymptotically best possible. The next chapter of the thesis then shows that a hamiltonian graph with minimal degree at least (5sqrt7)n/6 contains a 2factor with two components. The thesis then concludes with a chapter about X_H, which is the number of copies of a graph H in the random graph G(n,p). In particular, it will be shown that, for a connected graph H, the value of X_H modulo k is approximately uniformly distributed, provided that k is not too large a function of n.


Supervisor:  Scott, Alexander  Sponsor:  Engineering and Physical Sciences Research Council  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.559858  DOI:  Not available  
Keywords:  Combinatorics ; edgecolouring ; Ramsey theory ; random graphs  
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