Avoiding small subgraphs in Achlioptas processes

For a fixed integer r, consider the following random process. At each round, one is presentedwith r random edges from the edge set of the complete graph on n vertices, and is asked to chooseone of them. The selected edges are collected into a graph, which thus grows at the rate of oneedge per round. This is a natural generalization of what is known in the literature as an Achlioptasprocess (the original version has r = 2), which has been studied by many researchers, mainly in thecontext of delaying or accelerating the appearance of the giant component.In this paper, we investigate the small subgraph problem for Achlioptas processes. That is,given a fixed graph H , we study whether there is an online algorithm that substantially delaysor accelerates a typical appearance of H , compared to its threshold of appearance in the randomgraph G(n,M). It is easy to see that one cannot accelerate the appearance of any fixed graphby more than the constant factor r, so we concentrate on the task of avoiding H . We determinethresholds for the avoidance of all cycles Ct, cliques Kt, and complete bipartite graphs Kt,t, inevery Achlioptas process with parameter r ≥ 2.