Triangles in random graphs

Triangles in random graphs

Triangles in Random Graphs

OF THE DISSERTATION Triangles in Random Graphs by Robert DeMarco Dissertation Director: Jeff Kahn We prove four separate results. These results will appear or have appeared in various papers (see [10], [11], [12], [13]). For a gentler introduction to these results, the reader is directed to the first chapter of this thesis. Let G = Gn,p. With ξk = ξ n,p k the number of copies of Kk in G, p ≥ n−...

Online Ramsey games for triangles in random graphs

In the online F -avoidance edge-coloring game with r colors, a graph on n vertices is generated by at each stage randomly adding a new edge. The player must color each new edge as it appears; his goal is to avoid a monochromatic copy of F . Let N0(F, r, n) be the threshold function for the number of edges that the player is asymptotically almost surely able to paint before he loses. Even when F...

Directed triangles in directed graphs

We show that each directed graph (with no parallel arcs) on n vertices, each with indegree and outdegree at least n=t where t = 2:888997 : : : contains a directed circuit of length at most 3. c © 2003 Elsevier B.V. All rights reserved.

Triangles in H-free graphs

For two graphs T and H and for an integer n, let ex(n, T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T = K2 is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the case T = K3, which reveals several interesting phenomena. Three representativ...