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

Combinatorics;Discrete Mathematics