1,069 research outputs found
Positional games on random graphs
We introduce and study Maker/Breaker-type positional games on random graphs.
Our main concern is to determine the threshold probability for the
existence of Maker's strategy to claim a member of in the unbiased game
played on the edges of random graph , for various target families
of winning sets. More generally, for each probability above this threshold we
study the smallest bias such that Maker wins the biased game. We
investigate these functions for a number of basic games, like the connectivity
game, the perfect matching game, the clique game and the Hamiltonian cycle
game
Fast strategies in biased Maker--Breaker games
We study the biased Maker--Breaker positional games, played on the
edge set of the complete graph on vertices, . Given Breaker's bias
, possibly depending on , we determine the bounds for the minimal number
of moves, depending on , in which Maker can win in each of the two standard
graph games, the Perfect Matching game and the Hamilton Cycle game
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
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