Asymptotically Optimal Threshold Bias for the $(a : b)$ Maker-Breaker Minimum Degree, Connectivity and Hamiltonicity Games

TL;DR

This paper determines the asymptotically optimal bias thresholds for Maker in various $(a:b)$ Maker-Breaker games on complete graphs, including minimum degree, connectivity, and Hamiltonicity, resolving open problems and generalizing previous bounds.

Contribution

It introduces explicit winning strategies for Maker and Breaker in $(a:b)$ games, establishing asymptotically optimal bias bounds for spanning subgraphs, connectivity, and Hamiltonicity.

Findings

Derived asymptotic bias thresholds for Maker in minimum degree and connectivity games.

Provided explicit Breaker strategies matching Maker bounds, confirming optimality.

Resolved open problems on bias thresholds in Maker-Breaker games.

Abstract

We study the $(a:b)$ Maker-Breaker subgraph game played on the edges of the complete graph $K_n$ on $n$ vertices, $n,a,b \in \mathbb{N}$ where the goal of Maker is to build a copy of a specific fixed subgraph $H$. In our work this is a spanning graph with minimum degree $k=k(n)$, a connected spanning subgraph or a Hamiltonian subgraph. In the $(a:b)$ game in each round Maker chooses $a$ unclaimed edges of $K_n$ and Breaker chooses $b$ unclaimed edges. Maker wins, if he succeeds to build a copy of the subgraph under consideration, otherwise Breaker wins. For the $k$-minimum-degree, we present a winning strategy for Maker leading to a bound that generalizes a bound of Gebauer and Szab{\'o} for the $(1:b)$ case. Moreover, we give an explicit strategy for Breaker for $b >(1+o(1)) \frac{an}{a+\ln(n)}$ in case of $a=o\left(\sqrt{\frac{n}{\ln(n)}}\right)$ and $k=o(\ln(n))$. Note that this bound is the same as the Maker bound presented by Hefetz et al. (2012) for the $(a:b)$ connectivity game, which implies that the asymptotic optimal bias for this game is $\frac{an}{a+\ln(n)}$. This resolves the open problem stated by these authors. We also study the $(a:b)$ Hamiltonicity game in which Maker's goal is to create a Hamiltonian subgraph. For the $(1:b)$ variant Krivelevich proved that $\left(1+o(1) \right)\frac{n}{\ln n}$ is the exact threshold bias. Controlling Breaker's vertex degree in the $(a:b)$ Maker-Breaker minimum degree game enables us to the asymptotic optimal generalized threshold bias for the $(a:b)$-game, both for $a=o\left(\sqrt{\frac{n}{\ln n}} \right)$ and $a=\Omega\left(\sqrt{\frac{n}{\ln n}} \right)$.