Biased orientation games

TL;DR

This paper investigates biased orientation games on complete graphs, analyzing the minimum bias needed for Maker or Breaker to win in games involving cycles, Hamiltonicity, and fixed subgraph creation.

Contribution

It provides bounds on the bias thresholds for Maker and Breaker to win in three different orientation games involving cycles, Hamilton cycles, and fixed subgraph creation.

Findings

Bounds on bias for Maker to win cycle game

Bounds on bias for Maker to win Hamiltonicity game

Bounds on bias for Maker to create a fixed subgraph H

Abstract

We study biased {\em orientation games}, in which the board is the complete graph $K_n$, and Maker and Breaker take turns in directing previously undirected edges of $K_n$. At the end of the game, the obtained graph is a tournament. Maker wins if the tournament has some property $\mathcal P$ and Breaker wins otherwise. We provide bounds on the bias that is required for a Maker's win and for a Breaker's win in three different games. In the first game Maker wins if the obtained tournament has a cycle. The second game is Hamiltonicity, where Maker wins if the obtained tournament contains a Hamilton cycle. Finally, we consider the $H$-creation game, where Maker wins if the obtained tournament has a copy of some fixed graph $H$.