Structure-biased Maker-Breaker Games

TL;DR

The paper investigates structure-biased Maker-Breaker games on $K_n$, where Breaker must claim edge sets forming a fixed substructure per move. It derives order-of-magnitude threshold biases for three core goals (triangle, connectivity, Hamiltonicity) under three structures (clique $K_m$, matching $E_b$, and star $S_b$), revealing that such structural constraints sharply obstruct Breaker compared to classical games. Specifically, the thresholds are $m=\Theta(\sqrt{n})$ for the triangle under $K_m$, and $m=\Theta(\sqrt{n/\log n})$ for connectivity and Hamiltonicity under $K_m$, with all three goals under matching and star biases having $b=\Theta(n)$, and refined bounds for the star case ($0.1n \le b \le 3n/7$). These results show that enforcing structure on Breaker induces distinct strategic regimes and provide new insights for approaching classical Maker-Breaker problems.

Abstract

In classical Maker-Breaker games on graphs, Maker and Breaker take turns claiming edges; Maker's goal is to claim all of some structure (e.g., a spanning tree, Hamilton cycle, etc.), while Breaker aims to stop her. The standard question considered is how powerful a Breaker Maker can defeat; i.e., for the $(1:b)$-biased game where Breaker takes $b$ edges per turn, how large can $b$ be for Maker to still have a winning strategy, for various possible goal sets? We introduce a variant of this question in which Breaker is required to choose their multiple edges as the edges of (a subgraph of) a given structure (e.g., a matching, clique, etc.) on each turn. We establish the order of magnitude of the threshold biases for triangle games, connectivity games, and Hamiltonicity games under clique, matching, and star biases respectively. We conclude that in many cases structure imposes major obstruction to Breaker, opening up a set of games whose strategies deviate from the classical biased Maker-Breaker game strategies, and shedding light on the types of Breaker strategies that may or may not work to prove tighter bounds in the classical setting.

Theorems & Definitions (39)

• Theorem 1.1: Glazik and Srivastav glazik2022
• Theorem 1.2: Gebauer and Szabó gebauer2009
• Theorem 1.3: Krivelevich kriv2011
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 3.3