Connector-breaker games on random boards

Dennis Clemens, Laurin Kirsch, Yannick Mogge, TUHH Universitätsbibliothek
2021
The Maker-Breaker connectivity game on a complete graph K n or on a random graph G ∼ G n,p is well studied by now. Recently, London and Pluhár suggested a variant in which Maker always needs to choose her edges in such a way that her graph stays connected. It follows from their results that for this connected version of the game, the threshold bias on K n and the threshold probability on G ∼ G n,p for winning the game drastically differ from the corresponding values for the usual Maker-Breaker
more » ... ersion, assuming Maker's bias to be 1. However, they observed that the threshold biases of both versions played on K n are still of the same order if instead Maker is allowed to claim two edges in every round. Naturally, London and Pluhár then asked whether a similar phenomenon can be observed when a (2 : 2) game is played on G n,p . We prove that this is not the case, and determine the threshold probability for winning this game to be of size n −2/3+o(1) .
doi:10.15480/882.3667 fatcat:4j2kjg4yljdjdiz5ohvjlrlm2y