Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.813535 
Title:  Local limit theorem in random graphs and graphs on nonconstant surfaces  
Author:  Saller, Sophia 
ISNI:
0000 0004 9351 195X


Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  2020  
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Abstract:  
The thesis is split into two parts. In the first part we prove a local limit theorem for the number of appearances of the complete graph on four vertices, K_{4}, in the ErdösRényi Random graph G(n, p) for p in (0, 1) a fixed constant. The proof of this is based on bounding the characteristic function of the number of K_{4} in G(n, p) by using different conditioning arguments for different ranges of t. The second part treats classes of graphs embeddable in either an orientable surface of Euler genus at most g(n), denoted by G^(g(n)), or nonorientable surface of Euler genus at most g(n), denoted by H^(g(n)), for different nonconstant functions g(n). We first find bounds on the sizes of these classes and prove that as long as g(n) = o(n/log^3(n)), the classes G^(g(n)) and H^(g(n)) both have a growth constant equal to the planar graph growth constant. As long as g(n) = O(n/log(n)) it is further shown that the classes G^g(n) and H^(g(n)) are both small. We then go on to show which properties graphs in these graph classes commonly display, such as giving bounds on the number of edges, faces, pendant vertices, the maximum degree and the probability of connectedness.


Supervisor:  McDiarmid, Colin ; Riordan, Oliver  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.813535  DOI:  Not available  
Keywords:  Random graphs  
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