Doubly biased Walker-Breaker games

Abstract

We study doubly biased Walker--Breaker games, played on the edge set of a complete graph on $n$ vertices, $K_n$. Walker--Breaker game is a variant of Maker--Breaker game, where Walker, playing the role of Maker, must choose her edges according to a walk, while Breaker has no restrictions on choosing his edges.
Here we show that for $b\leq \frac{n}{10\ln{n}}$, playing a $(2:b)$ game on $E(K_n)$, Walker can create a graph containing a spanning tree.
Also, we determine a constant $c > 0$ such that Walker has a strategy to make a Hamilton cycle of $K_n$ in the $(2 : \frac{cn}{\ln{n}})$ game.