Avoiding small subgraphs in Achlioptas processes

TL;DR

This paper studies how to delay or accelerate the appearance of small subgraphs like cycles, cliques, and bipartite graphs in generalized Achlioptas processes, revealing thresholds for avoidance in these random graph processes.

Contribution

It establishes thresholds for avoiding specific small subgraphs in Achlioptas processes, extending understanding beyond the classical case.

Findings

Thresholds for avoiding cycles C_t identified

Thresholds for avoiding cliques K_t determined

Thresholds for avoiding bipartite graphs K_{t,t} established

Abstract

For a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r=2), which has been studied by many researchers, mainly in the context 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 delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n, M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles C_t, cliques K_t, and complete bipartite graphs K_{t,t}, in every Achlioptas process with parameter r >= 2.