A Constructive Winning Maker Strategy in the Maker-Breaker $C_4$-Game

TL;DR

This paper presents the first explicit strategy for Maker to win the $C_4$-game in a Maker-Breaker setting, achieving a winning threshold close to the known asymptotic limit with a significantly improved constant.

Contribution

The authors develop the first constructive winning strategy for Maker in the $C_4$-Maker-Breaker game, improving the constant factor for the winning threshold.

Findings

Maker can win if q < 0.16 n^{2/3}

The strategy is constructive and explicit

Threshold is close to the asymptotic limit

Abstract

Maker-Breaker subgraph games are among the most famous combinatorial games. For given $n,q \in \mathbb{N}$ and a subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. In each round of the game Maker claims one edge and Breaker is allowed to claim up to $q$ edges. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. In this work we introduce the first constructive strategy for Maker for the $C_4$-Maker-Breaker game and show that he can win the game if $q < 0.16 n^{2/3}$. According to the theorem of Bednarska and Luczak (2000) $n^{2/3}$ is asymptotically optimal for this game, but the constant given there for a random Maker strategy is magnitudes apart from our constant 0.16.