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

Abstract: 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 Łuczak (2000) determined in a landmark work the asymptotics of the treshold bias as $Θ(n^{1/m(C)})$ 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 $Θ$-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 Łuczak (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.

• The paper introduces the first constructive, polynomial-time winning strategy for Breaker in Maker-Breaker Ck-games for all fixed k ≥ 4.
• It employs a sophisticated potential function method to claim obligatory edges and control the cumulative potential during biased play.
• The strategy achieves asymptotically optimal bias bounds with exponentially tighter constants compared to previous probabilistic strategies.

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

Introduction and Context

The Maker-Breaker $C_k$-game is a central object in positional game theory, played on the edge set of the complete graph $K_n$. Two players, Maker and Breaker, alternately claim edges, with Maker taking one per turn and Breaker up to $q$ edges—the bias parameter. Maker’s goal is to occupy all edges of a copy of a fixed subgraph $C$ (here, a $k$-cycle), while Breaker aims to prevent this outcome.

The threshold bias $q_C$, where the advantage shifts from Maker to Breaker, asymptotically follows $\Theta(n^{1/m(C)})$, with $m(C)$ the maximum $2$-density over non-trivial subgraphs, per Bednarska and Łuczak [Combinatorica 2000]. However, obtaining sharp constants—especially via explicit strategies rather than probabilistic existence proofs—has remained elusive for fixed cycles $C_k$0, $C_k$1.

The triangle ($C_k$2) case is extensively studied: Chvátal and Erdős established that Maker wins for $C_k$3, Breaker for $C_k$4. Glazik and Srivastav improved the constructive bias upper bound for Breaker to $C_k$5 using a potential function method [EJC 2022], nearly matching the lower bound. Generalizing such constructive methods to $C_k$6, $C_k$7, especially with polynomial-time strategies, is a prominently open question.

Main Contributions

The paper rigorously develops a general constructive, polynomial-time strategy for Breaker in biased Maker-Breaker $C_k$8-games for arbitrary fixed $C_k$9. The approach synthesizes and generalizes the potential function strategy introduced for triangles to cycles of arbitrary fixed length, confirming conjectures by Joel Spencer on the applicability of such methods beyond $K_n$0.

Potential Function Framework

The core innovation is the systematic formalization of a “potential function” strategy for Breaker, which operates generically for arbitrary fixed winning subgraphs $K_n$1—here, focus is on cycles. The strategy prioritizes “obligatory” edges (those that, if neglected, would allow Maker to complete $K_n$2 in the next move), and otherwise claims edges maximizing a carefully constructed vertex-based potential.

For the $K_n$3-game, the constructive bias bound is established:

$K_n$4

is sufficient for Breaker's win via an explicit deterministic strategy.

Crucially, the constants in this bound are exponentially tighter in $K_n$5 compared to those derived from prior random strategies (such as the constants in Bednarska and Łuczak), resulting in genuine improvements. All strategy steps are directly implementable in polynomial time.

Strategy Formulation and Analysis

The strategy relies on several algorithmic and analytical tools:

• Potential Function Construction: The potential at each vertex is parameterized by the difference between Maker and Breaker degrees, mapped through an exponential function with carefully tuned parameters (the “balance function”). This leverages the structural dependencies of cycles.
• Edge Selection Procedure: In each Breaker turn, all currently “obligatory” edges are claimed first. Then, up to $K_n$6 total claims are performed by iteratively selecting unclaimed edges of highest potential.
• Threshold Rigor: A detailed potential analysis proves that as long as Breaker’s bias $K_n$7 surpasses the stated threshold, the cumulative potential (over all vertices) can never cross $K_n$8 during the game, thereby ensuring Maker cannot complete $K_n$9. This is shown via recursive bounds on the potential-increasing effect of Maker’s moves and the potential-lowering effect of Breaker’s bias.

The argument includes careful attention to edge-case analysis (e.g., obligatory edges, maximum degrees), and iterative bounding using parameters that decay with $q$0, rendering the bounds asymptotically optimal as $q$1. The proof is constructive and the update mechanisms for degrees and potentials are fully explicit.

Comparison to Previous Work

$q$2	Maker lower bound (best known, constructive)	Constructive Breaker upper bound (this paper)
3	$q$3 [Chvátal-Erdős]	$q$4 [Glazik, Srivastav]
4	$q$5 [Sowa, Srivastav]	$q$6 (this paper)
5	$q$7 [Bednarska, Łuczak]	$q$8 (this paper)
6	$q$9 [Sowa, PhD]	$C$0 (this paper)

This demonstrates significant improvement in constructively attainable constants for Breaker, and further closes the gap with the best existential results for Maker.

Implications and Future Directions

Theoretical Significance:

The potential function method demonstrated here marks a decisive step in bridging the gap between existential (usually probabilistic) and algorithmic (constructive, polynomial-time) results in biased positional games on graphs. The framework is modular and potentially extensible to broader classes of winning subgraphs, notably towards cliques such as $C$1, which remain a compelling further target.

Algorithmic Impact:

The explicit nature of the strategy, with $C$2 worst-case complexity, makes it viable as not merely a theoretical tool but as a directly implementable algorithm for combinatorial game solvers. The strict polynomiality ensures practical applicability for moderate $C$3 and fixed $C$4.

Future Work:

Key avenues for extension include:

• Designing concrete potential functions for other fixed subgraphs such as $C$5;
• Developing corresponding constructive Maker strategies, especially for $C$6, $C$7, towards closing the residual constant gaps;
• Extending constructive results to the scenario where the winning subgraph $C$8 depends on $C$9 (e.g., Hamiltonicity, connectivity);
• Pursuing derandomization of random strategies for both players in the spirit of Bednarska and Łuczak, potentially yielding polynomial-time deterministic protocols in broader classes of positional games.

Conclusion

The paper provides the first constructive, polynomial-time winning strategies for Breaker in the Maker-Breaker $k$0-cycle game for all fixed $k$1, with bias bounds that are asymptotically optimal and constants strictly improving on prior probabilistic approaches. The generalization of the potential function method to arbitrary fixed cycles resolves longstanding open conjectures and offers a versatile paradigm for attacking further biased positional games. This significantly advances the algorithmic combinatorics of positional games and lays the groundwork for further closing the gap between existential and constructive thresholds in subgraph games.

Reference:

"Constructive Winning Breaker Strategies in the Maker-Breaker $k$2-Game" (2607.01294)