Creating cycles in Walker-Breaker games

TL;DR

This paper investigates the capabilities of a constrained Walker in biased Walker-Breaker games to create cycles of specific lengths, analyzing how bias affects Walker's success in forming such cycles.

Contribution

It introduces a detailed analysis of cycle creation in biased Walker-Breaker games, addressing open questions about Walker's potential to form cycles of fixed lengths under various biases.

Findings

Walker can create cycles of certain lengths depending on bias

Threshold bias levels for cycle formation are identified

Strategies for Walker to maximize cycle length are proposed

Abstract

We consider biased $(1:b)$ Walker-Breaker games: Walker and Breaker alternately claim edges of the complete graph $K_n$, Walker taking one edge and Breaker claiming $b$ edges in each round, with the constraint that Walker needs to choose her edges according to a walk. As questioned in a paper by Espig, Frieze, Krivelevich and Pegden, we study how long a cycle Walker is able to create and for which biases $b$ Walker has a chance to create a cycle of given constant length.