Small subgraphs in random graphs and the power of multiple choices

Abstract

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 G 0 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 G N . In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in G N 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 N 0 ( F , r , n ) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1 − o ( 1 ) as n → ∞ ) avoids creating a copy of F for as long as N = o ( N 0 ) , and prove that any online strategy will a.a.s. create such a copy once N = ω ( N 0 ) .