Linear Bounds for Cycle-Free Saturation Games

Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with only constraint being that neither player can add edge creates subgraph in $\mathcal{F}$. The ends when no more be added graph. One wishes end quickly possible, while other prolong game. We let $\textrm{sat}_g(n,\mathcal{F})$ denote number are final both play optimally.In general there very few non-trivial bounds order magnitude $\textrm{sat}_g(n,\mathcal{F})$. In this work, find collections infinite families cycles $\mathcal{C}$ such $\textrm{sat}_g(n,\mathcal{C})$ has linear growth rate.