# Spanning Structures in Walker--Breaker Games

## TL;DR

This paper investigates biased Walker--Breaker games on complete graphs, demonstrating Walker's ability to win connectivity and Hamilton cycle games against Breaker with bias proportional to n/ln n, highlighting strategic advantages despite restrictions.

## Contribution

It introduces the Walker--Breaker game variant and proves Walker can win key graph games against significant bias, extending understanding of positional game strategies.

## Key findings

- Walker can win connectivity games with bias up to order n/ln n.
- Walker can win Hamilton cycle games with similar bias.
- The results extend the theory of biased positional games.

## Abstract

We study the biased $(2:b)$ Walker--Breaker games, played on the edge set of the complete graph on $n$ vertices, $K_n$. These games are a variant of the Maker--Breaker games with the restriction that Walker (playing the role of Maker) has to choose her edges according to a walk. We look at the two standard graph games -- the Connectivity game and the Hamilton Cycle game and show that Walker can win both games even when playing against Breaker whose bias is of the order of magnitude $n/ \ln n$.