Zhen HE, Mei LU
Let $F$, $G$ and $H$ be three graphs with $G\subseteq{H}$. We call $G$ an $F$-saturated graph relative to $H$, if there is no copy of $F$ in $G$ but there is a copy of $F$ in $G+e$ for any $e\in E(H)\setminus E(G)$. The $F$-saturation game on host graph $H$ consists of two players, named Max and Min, who alternately add edges of $H$ to $G$ such that each chosen edge avoids creating a copy of $F$ in $G$, and the players continue to choose edges until $G$ becomes $F$-saturated relative to $H$. Max wishes to maximize the length of the game, while Min wishes to minimize the process. Let ${\rm sat}_g(F,H)$ (resp. ${\rm sat}_{g}^{'}(F,H)$) denote the number of edges chosen when Max (resp. when Min) starts the game and both players play optimally. In this article, we show that ${\rm sat}_g(P_5,K_n) = {\rm sat}_g^{'}(P_5,K_n)= n+2$ for $n\ge 15$, and ${\rm sat}_g(P_5,K_{m,n})$, ${\rm sat}_g^{'}(P_5,K_{m,n})$ lie in $\{m+n-\lfloor \frac{m+2}{4}\rfloor, m+n-\lceil \frac{m-3}{4}\rceil \}$ if $n\ge\frac{5}{2}m$ and $m\ge 4$, respectively.
