Gal Kronenberg

Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

We consider random-turn positional games, introduced by Peres, Schramm, Sheeld and Wilson in 2007. A p-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is p). We analyze the random-turn version of several classical MakerBreaker games such as the game Box (introduced by Chv atal and Erd} os in 1987), the Hamilton cycle game and the k-vertex-connectivity game (both played on the edge set of Kn). For each of these games we provide each of the players with a (randomized) ecient strategy which typically ensures his win in the asymptotic order of the minimum value of p for which he typically wins the game, assuming optimal strategies of both players.

Packing, Counting and Covering Hamilton cycles in random directed graphs

Abstract A Hamilton cycle in a digraph is a cycle passing through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Posa ‘rotation-extension’ technique for the undirected analogue. Here, we present a general and a very simple method, using known results, to attack problems of packing, counting and covering Hamilton cycles in random directed graphs, for every edge-probability p > log C ⁡ ( n ) / n . Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case.

On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of

Packing a randomly edge-colored random graph with rainbow

G\sim G\left(n,\frac {1+\varepsilon}n\right)

k

in terms of

-outs

\varepsilon

Optimal Threshold for a Random Graph to be 2-Universal

. We then apply this result to prove the following conjecture by Frieze and Pegden. For every

Random-Player Maker-Breaker games

\varepsilon>0

Goldberg's Conjecture is True for Random Multigraphs

there exists

Semi-random graph process

\ell_\varepsilon