How fast can Maker win in fair biased games?

TL;DR

This paper investigates the minimum number of moves Maker requires to win various positional games on complete graphs and explores Red's winning strategies in strong game variants.

Contribution

It provides new bounds on Maker's winning move counts and characterizes Red's winning strategies in the strong versions of these games.

Findings

Determined the minimum moves for Maker to win in four different games.

Established Red's winning strategies in the strong game variants.

Provided bounds and strategies for both Maker and Red in these graph games.

Abstract

We study (a:a) Maker-Breaker games played on the edge set of the complete graph on n vertices. In the following four games - perfect matching game, Hamilton cycle game, star factor game and path factor game, our goal is to determine the least number of moves which Maker needs in order to win these games. Moreover, for all games except for the star factor game, we show how Red can win in the strong version of these games.