# Connector-Breaker games on random boards

### Working Groups: Chair Discrete Mathematics

### Collaborators (MAT): Dr.Â Dennis Clemens, Yannick Mogge, M. Sc.

### Collaborators (External): Laurin Kirsch

## Description

By now, the Maker-Breaker connectivity game on a complete graph \(K_n\) or on a random graph \(G\sim G_{n,p}\) is well studied. Recently, London and PluhĂˇr suggested a variant in which Maker always needs to choose her edges in such a way that her graph stays connected. By their results it follows that for this connected version of the game, the threshold bias on \(K_n\) and the threshold probability on \(G\sim G_{n,p}\) for winning the game drastically differ from the corresponding values for the usual Maker-Breaker version, assuming Makerâ€™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, this made London and PluhĂˇr ask 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 \(n^{-2/3+o(1)}\).