Connector-Breaker games on random boards

TL;DR

This paper investigates a variant of the Maker-Breaker connectivity game called the connector-breaker game on random graphs, establishing the threshold probability for Maker's win when claiming two edges per turn.

Contribution

The paper determines the threshold probability for Maker's win in the connector-breaker game played on random graphs with a (2:2) claiming scheme, extending previous results.

Findings

Threshold probability for Maker's win is approximately n^{-2/3+o(1)}.

The threshold bias behavior differs from the standard Maker-Breaker game.

The (2:2) game on G_{n,p} exhibits a distinct phase transition threshold.

Abstract

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 Pluh\'ar 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 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, this made London and Pluh\'ar 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 $n^{-2/3+o(1)}$.