Small subgraphs in random graphs and the power of multiple choices

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G0 as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r − 1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph GN . In a recent paper by Krivelevich, Loh, and Sudakov [11], the problem of avoiding a copy of some fixed graph F in GN for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F. In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N0(F, r, n) for arbitrary graphs F and any fixed i...

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Torsten Mütze, Reto Spöhel, Henning Thomas
• Keywords: Arbitrary Graphs, Explicit Threshold Functions, General Threshold Formula, JCT 2011