Constructive Winning Breaker Strategies in the Maker-Breaker $C_k$-Game

Maker-Breaker subgraph games are among the most famous combinatorial games. For $n,q\in\mathbb{N}$ and a fixed subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. Maker claims one unclaimed edge per round and Breaker may claim up to $q$ edges per round. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. Bednarska and {\L}uczak (2000) determined in a landmark work the asymptotics of the treshold bias as $\Theta(n^{1/m(C)})$ where $m(C)$ is the 2-density of $C$, analysing random strategies. Since then it has been a major open problem to determine the treshhold bias, if it exists, with corresponding strategies, leading to sharp constants in the $\Theta$-notion. A famous case is the triangle game ($C=C_3$), studied by Chvatal and Erd"os (1978), who showed Maker wins if $q\le \sqrt{2n}$ and Breaker wins if $q\ge2\sqrt{n}$. Glazik and Srivastav (2022) improved this via a potential method, showing Breaker wins already for $q\ge\sqrt{8/3}\sqrt{n}$. Spencer (2019) conjectured generalizability to arbitrary subgraphs $C$. We confirm this conjecture, presenting a general winning strategy for Breaker if the potential function fullfils conditions depending on $C$. With this result we give the first constructive (polynomial-time) strategies for Breaker in the $k$-cycle Maker-Breaker game for arbitrary, but fixed $k \geq 4$: Breaker wins if $q>\sqrt[k-1]{(k-1)\big(\frac{2(k-1)}{k}\big)^{k-2}n^{k-2}}$. By Bednarska and {\L}uczak (2000) our bound is asymptotically optimal. However, our constants are better than those arising from their random strategies. More recently, Sowa and Srivastav (2025) gave the first constructive Maker strategy for $C_4$. Our work may motivate study of Maker strategies for $C_k, k \ge 5$, narrowing the gap towards the Breaker bounds presented.