Random maximal H-free graphs

Abstract

Given a graph H, a random maximal H-free graph is constructed by the following random greedy process. First assign to each edge of the complete graph on n vertices a birthtime which is uniformly distributed in [0, 1]. At time p=0 start with the empty graph and increase p gradually. Each time a new edge is born, it is included in the graph if this does not create a copy of H. The question is then how many edges such a graph will have when p=1. Here we give asymptotically almost sure bounds on the number of edges if H is a strictly 2-balanced graph, which includes the case when H is a complete graph or a cycle. Furthermore, we prove the existence of graphs with girth greater than and chromatic number n*y1/(-1)+o(1), which improves on previous bounds for >3. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 61–82, 2001