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 ).

Offline thresholds for Ramsey‐type games on random graphs

“…This is needed in order to obtain an offline version of the problem with a well-defined distribution ( , , in our case). As argued in [11], the threshold of the online game does not depend on the precise definition used.…”

“…The Achlioptas game was first investigated in [5], and solved completely in [11], where a general threshold formula 0 ( , , ) valid for all graphs and every integer ≥ 2 was determined. This formula is rather complicated and contains both a minimization over all possible orders in which the edges of may appear, and a maximization over entire sequences of appropriate subgraphs of .…”

“…The motivation for this work comes from three Ramsey-type one-player games that have been studied in an online setting in previous work: the online Ramsey game [2,9,10], the balanced online Ramsey game [7,12], and the Achlioptas game [5,11]. In all three games, the edges of the complete graph appear in a random order, either one by one or in batches of some fixed size.…”

Creating Small Subgraphs in Achlioptas Processes With Growing Parameter

“…Only recently, other properties have been studied: the problem of accelerating Hamiltonicity in Achlioptas processes was investigated in [14], and the problem of delaying the occurrence of a given fixed graph as a subgraph was studied in [13,19]. In this work we are concerned with the opposite problem of the latter; i.e., throughout this paper our goal is to create a copy of some given fixed graph H as quickly as possible.…”

“…For fixed values of r, the general threshold function of this problem was determined in [19], but for the case where r grows the question is wide open.…”