The Speed and Threshold of the Biased Hamilton Cycle Game

TL;DR

This paper analyzes the biased Hamilton cycle game, establishing bounds on the speed and threshold for Maker's victory in terms of game parameters and providing precise winning conditions.

Contribution

It introduces new bounds on the threshold bias and the number of steps for Maker to win the biased Hamilton cycle game, advancing understanding of game dynamics.

Findings

Maker wins within a specific step bound for certain bias thresholds.

The threshold bias is characterized as proportional to n/ln(n).

The number of steps for Maker's victory is explicitly bounded.

Abstract

We show that there is a constant C such that for any $b<\frac{n}{\ln{n}}-\frac{Cn}{(\ln{n})^{3/2}}$, Maker wins the Maker-Breaker Hamilton cycle game in $n+\frac{Cn}{\sqrt{\ln{n}}}$ steps.