# On the WalkerMaker-WalkerBreaker games

## TL;DR

This paper investigates WalkerMaker-WalkerBreaker games on complete graphs, focusing on connectivity and Hamilton cycle games, and demonstrates rapid winning strategies for WalkerMaker under walk-based constraints.

## Contribution

It introduces a new variant of Maker-Breaker games with walk restrictions and analyzes winning strategies for WalkerMaker in connectivity and Hamilton cycle scenarios.

## Key findings

- WalkerMaker can win both games quickly under walk constraints
- The study provides bounds on the winning time for WalkerMaker
- Walk-based restrictions significantly influence game dynamics

## Abstract

We study the unbiased WalkerMaker-WalkerBreaker games on the edge set of the complete graph on $n$ vertices, $K_n$, a variant of well-known Maker-Breaker positional games, where both players have the restriction on the way of playing. Namely, each player has to choose her/his edges according to a walk. Here, we focus on two standard graph games - the Connectivity game and the Hamilton cycle game and show how quickly WalkerMaker can win both games.