Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Lower Bounds for the Size of Random Maximal H-Free Graphs. | Lower Bounds for the Size of Random Maximal H-Free Graphs Guy Wolfovitz Department of Computer Science Haifa University Haifa Israel gwolfovi@ Submitted Jun 19 2008 Accepted Dec 15 2008 Published Jan 7 2009 Mathematics Subject Classihcations 05C80 05C35 Abstract We consider the next random process for generating a maximal H-free graph Given a Hxed graph H and an integer n start by taking a uniformly random permutation of the edges of the complete n-vertex graph Kn. Then traverse the edges of Kn according to the order imposed by the permutation and add each traversed edge to an initially empty evolving n-vertex graph - unless its addition creates a copy of H. The result of this process is a maximal H-free graph Mn H . Our main result is a new lower bound on the expected number of edges in Mn H for H that is regular strictly 2-balanced. As a corollary we obtain new lower bounds for Turán numbers of complete balanced bipartite graphs. Namely for Hxed r 5 we show that ex n Kr r Q n2-2 r 1 lnlnn 1 r -1 . This improves an old lower bound of Erdos and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in Mn H conditioned on the event that the edge is traversed relatively but not trivially early during the process. 1 Introduction Consider the next random process for generating a maximal H-free graph. Given n 2 N and a graph H assign every edge f of the complete n-vertex graph Kn a birthtime p f distributed uniformly at random in the interval 0 1 . Note that with probability 1 the birthtimes are distinct and so p is a permutation. Now start with the empty n-vertex graph and iteratively add edges to it as follows. Traverse the edges of Kn in order of their birthtimes starting with the edge whose birthtime is smallest and add each traversed edge to the evolving graph unless its addition creates a copy of H. When all edges of Kn have been exhausted the process ends. Denote by Mn H the graph which is the
