Graph explorer

On saturation games

A graph $G = (V,E)$ is said to be saturated with respect to a monotone increasing graph property ${\mathcal P}$, if $G \notin {\mathcal P}$ but $G \cup \{e\} \in {\mathcal P}$ for every $e \in \binom{V}{2} \setminus E$. The saturation game $(n, {\mathcal P})$ is played as follows. Two players, called Mini and Max, progressively build a graph $G \subseteq K_n$, which does not satisfy ${\mathcal P}$. Starting with the empty graph on $n$ vertices, the two players take turns adding edges $e \in \binom{V(K_n)}{2} \setminus E(G)$, for which $G \cup \{e\} \notin {\mathcal P}$, until no such edge exists (i.e. until $G$ becomes ${\mathcal P}$-saturated), at which point the game is over. Max's goal is to maximize the length of the game, whereas Mini aims to minimize it. The score of the game, denoted by $s(n, {\mathcal P})$, is the number of edges in $G$ at the end of the game, assuming both players follow their optimal strategies. We prove lower and upper bounds on the score of games in which the property the players need to avoid is being $k$-connected, having chromatic number at least $k$, and admitting a matching of a given size. In doing so we demonstrate that the score of certain g