Maker Breaker on Digraphs

Abstract

We study two biassed Maker-Breaker games played on the complete digraph ${\vec{K}}_n$. In the strong connectivity game, Maker wants to build a strongly connected subgraph. We determine the asymptotic optimal bias for this game viz. $\frac{n}{\log n}$. In the Hamiltonian game, Maker wants to build a Hamiltonian subgraph. We determine the asymptotic optimal bias for this game up to a constant factor.

We clarify an important argument in the appendix