Asymptotic Properties Of Random Graphs

Author : Graeme Kemkes
Publisher :
ISBN 13 :
Total Pages : 86 pages
Book Rating : 4.:/5 (613 download)

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Download or read book Properties of Random Graphs written by Graeme Kemkes and published by . This book was released on 2008 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: The thesis describes new results for several problems in random graph theory. The first problem relates to the uniform random graph model in the supercritical phase; i.e. a graph, uniformly distributed, on n vertices and M=n/2+s edges for s=s(n) satisfying n2/3=o(s) and s=o(n). The property studied is the length of the longest cycle in the graph. We give a new upper bound, which holds asymptotically almost surely, on this length. As part of our proof we establish a result about the heaviest cycle in a certain randomly-edge-weighted nearly-3-regular graph, which may be of independent interest. Our second result is a new contiguity result for a random d-regular graph. Let j=j(n) be a function that is linear in n. A (d, d-1)-irregular graph is a graph which is d-regular except for 2j vertices of degree d-1. A j-edge matching in a graph is a set of j independent edges. In this thesis we prove the new result that a random (d, d-1)-irregular graph plus a random j-edge matching is contiguous to a random d-regular graph, in the sense that in the two spaces, the same events have probability approaching 1 as n [arrr] [infin]. This allows one to deduce properties, such as colourability, of the random irregular graph from the corresponding properties of the random regular one. The proof applies the small subgraph conditioning method to the number of j-edge matchings in a random d-regular graph. The third problem is about the 3-colourability of a random 5-regular graph. Call a colouring balanced if the number of vertices of each colour is equal, and locally rainbow if every vertex is adjacent to vertices of all the other colours. Using the small subgraph conditioning method, we give a condition on the variance of the number of locally rainbow balanced 3-colourings which, if satisfied, establishes that the chromatic number of the random 5-regular graph is asymptotically almost surely equal to 3. We also describe related work which provides evidence that the condition is likely to be true. The fourth problem is about the chromatic number of a random d-regular graph for fixed d. Achlioptas and Moore recently announced a proof that a random d-regular graph asymptotically almost surely has chromatic number k-1, k, or k+1, where k is the smallest integer satisfying d