Combinatorica
Avoidance of a giant component in half the edge set of a random graph
Random Structures & Algorithms
Combinatorics, Probability and Computing
Random Structures & Algorithms
Online balanced graph avoidance games
European Journal of Combinatorics
Combinatorica
Avoiding small subgraphs in Achlioptas processes
Random Structures & Algorithms - Proceedings of the Thirteenth International Conference “Random Structures and Algorithms” held May 28–June 1, 2007, Tel Aviv, Israel
Online ramsey games in random graphs
Combinatorics, Probability and Computing
Offline thresholds for Ramsey-type games on random graphs
Random Structures & Algorithms - Special 20th Anniversary Issue
Hamiltonicity thresholds in Achlioptas processes
Random Structures & Algorithms
Coloring random graphs online without creating monochromatic subgraphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On balanced coloring games in random graphs
European Journal of Combinatorics
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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=@w(N"0).