A One-Third Bound for the Maker--Breaker Degree Game

Kaizhe Chen; Xiaonan Chen; Haotian Yang

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arXiv · 2608.20644

A One-Third Bound for the Maker--Breaker Degree Game

Kaizhe Chen·Xiaonan Chen·Haotian Yang

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Abstract

Let $H$ be a finite hypergraph with rank $r$ and minimum degree $d$. In the Maker--Breaker degree game, Maker and Breaker alternately claim previously unclaimed hyperedges of $H$, with Maker moving first, and Breaker seeks to maximize the minimum degree $δ(H_{\mathrm B})$ of his spanning subhypergraph $H_{\mathrm B}$. We prove that, for every fixed $r\ge2$ and all sufficiently large $d$, Breaker has a deterministic strategy satisfying \[ δ(H_{\mathrm B})\ge \frac{d}{r+1}-\sqrt{d\log d}. \] For graphs, this improves the classical universal lower bound $d/4$ to $d/3-\sqrt{d\log d}$. The proof introduces a virtual balancing process, encodes local imbalance by a multiplicative risk, and keeps the resulting risk vector in the Shearer region throughout the game.

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Kaizhe Chen, Xiaonan Chen, Haotian Yang. 2026-08-21. A One-Third Bound for the Maker--Breaker Degree Game. https://arxiv.org/abs/2608.20644

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