Fast strategies in biased Maker--Breaker games

TL;DR

This paper analyzes biased Maker--Breaker games on complete graphs, establishing bounds on the minimal moves Maker needs to win in perfect matching and Hamilton cycle games depending on Breaker's bias.

Contribution

It provides new bounds on the minimal number of moves for Maker to win in biased games on complete graphs, considering the bias's dependence on the number of vertices.

Findings

Bounds for Maker's minimal winning moves in Perfect Matching game.

Bounds for Maker's minimal winning moves in Hamilton Cycle game.

Results depend on Breaker's bias $b$ and the number of vertices $n$.

Abstract

We study the biased $(1:b)$ Maker--Breaker positional games, played on the edge set of the complete graph on $n$ vertices, $K_n$. Given Breaker's bias $b$, possibly depending on $n$, we determine the bounds for the minimal number of moves, depending on $b$, in which Maker can win in each of the two standard graph games, the Perfect Matching game and the Hamilton Cycle game.