Maker-Breaker-Crossing-Game on the Triangular Grid-graph

We study the $(p,q)$-Maker Breaker Crossing game introduced by Day and Falgas Ravry in 'Maker-Breaker percolation games I: crossing grids'. The game described in their paper involves two players Maker and Breaker who take turns claiming p and q as yet unclaimed edges of the graph respectively. Maker aims to make a horizontal path from a leftmost vertex to a rightmost vertex and Breaker aims to prevent this. The game is a version of the more general Shannon switching game and is played on a square grid graph. We consider the same game played on the triangular grid graph $\Delta_{(m,n)}$ (m vertices across, n vertices high) and aim to find, for given $(p,q,m,n)$, a winning strategy for Maker or Breaker. We establish using a similar strategy to that used by Day and Falgas Ravry to show that: $\bullet$ For sufficiently tall grids and $p\geq q$ Maker has a winning strategy for the $(p,q)$-crossing game on $\Delta_{(m,n)}$ . $\bullet$ For sufficiently wide grids and $4p\leq q$, Breaker has a winning strategy for the $(p,q)$-crossing game on $\Delta_{(m,n)}$.